Precalculus with Limits 3rd Edition Larson Solutions Manual by Shay

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NOT FOR SALE CONTENTS

Part I

Solutions to All Exercises..................................................................... 1

Chapter 1

Functions and Their Graphs................................................................... 1

Chapter 2

Polynomial and Rational Functions...................................................118

Chapter 3

Exponential and Logarithmic Functions ...........................................236

Chapter 4

Trigonometry......................................................................................301

Chapter 5

Analytic Trigonometry.......................................................................392

Chapter 6

Additional Topics in Trigonometry ...................................................464

Chapter 7

Systems of Equations and Inequalities ..............................................542

Chapter 8

Matrices and Determinants ................................................................635

Chapter 9

Sequences, Series, and Probability ....................................................728

Chapter 10

Topics in Analytic Geometry.............................................................800

Chapter 11

Analytic Geometry in Three Dimensions..........................................935

Chapter 12

Limits and an Introduction to Calculus .............................................979

Appendix A

Review of Fundamental Concepts of Algebra ................................1039 Solutions to Checkpoints ...............................................................1085 Solutions to Practice Tests.............................................................1290

Part II

Solutions to Chapter and Cumulative Tests ...............................1316

This sample only, Download all chapters at: alibabadownload.com

NOT FOR SALE

NOT FOR SALE C H A P T E R 1 Functions and Their Graphs Section 1.1

Rectangular Coordinates ........................................................................2

Section 1.2

Graphs of Equations ...............................................................................8

Section 1.3

Linear Equations in Two Variables .....................................................18

Section 1.4

Functions...............................................................................................31

Section 1.5

Analyzing Graphs of Functions ...........................................................40

Section 1.6

A Library of Parent Functions .............................................................51

Section 1.7

Transformations of Functions ..............................................................55

Section 1.8

Combinations of Functions: Composite Functions.............................66

Section 1.9

Inverse Functions..................................................................................75

Section 1.10

Mathematical Modeling and Variation................................................88

Review Exercises ..........................................................................................................95 Problem Solving .........................................................................................................110 Practice Test .............................................................................................................116

NOT FOR SALE C H A P T E R 1 Functions and Their Graphs Section 1.1 Rectangular Coordinates 1. Cartesian

15.

Year, x Number of Stores, y

2. Origin; quadrants

2003

4906

3. Distance Formula

2004

5289

4. Midpoint Formula

2005

6141

2006

6779

2007

7262

2008

7720

2009

8416

2010

8970

Year t 16.

7. 3, 4

Month, x Temperature, y

8. 12, 0

– 39

9. x ! 0 and y 0 in Quadrant IV.

– 39

10. x 0 and y 0 in Quadrant III.

– 29

4 and y ! 0 in Quadrant II.

–5

– 23

– 34

12. y 5 in Quadrant III or IV. 13.

x, y

is in the second Quadrant means that x, y is

in Quadrant III. 14.

x, y , xy

! 0 means x and y have the same signs.

This occurs in Quadrant I or III.

Temperature (in °F)

11. x

3 l 2003

30 20 10 x

0 −10

10 12

−20 −30 −40

Month (1 ↔ January)

INSTRUCTOR USE ONLY 2

NOT FOR SALE Section ection 1.1

17. d

x1 y2 y1

3 2

6 6

22. d

0 8

20 5

23. (a) 1, 0 , 13, 5

1 4

3 1

Distance

25 144

169

132

The distance between 9, 1 and 9, 4 is 3. The distance between 1, 1 and 9, 4 is 2 3

9 1

(b) 102 32

x1 y2 y1

1Â· 4Â· Â§ Â§ Â¨ 2 Â¸ Â¨ 1 Â¸ 2Â¹ 3Â¹ Â© Â© 2

Â§ 3Â· Â§ 7Â· Â¨ Â¸ Â¨ Â¸ 2 Â© Â¹ Â© 3Â¹ 9 49 4 9 277 36

4 1

109

100 9

109

4 2 2

0 1

4 1

4 1 2

0 5

25 25

2 1

1 5

5 2

3 1

26. d1 d2

5 3

5 1

9 36

50 45

16 4 4 16

1 3

5 3

1 5

109.

25. d1

29 units

1 13

24. (a) The distance between 1, 1 and 9, 1 is 10.

4 25

21. d

(b) 52 122

x1 y2 y1

5 0

61 units

169

1, 0 , 13, 0

36 25

20. d

13, 5 , 13, 0

Distance

5 1

5 0

122 52

x1 y2 y1

13 1 2

Distance

17 units

289

| 17.21 units

64 225

19. d

296.2

x1 y2 y1

9.5 8.2 2.6

10.8

179.56 116.64

25 144

13.4 2

13 units 18. d

x1 y2 y1

3.9

Rectangular Coordinates

36 4

20 20 40

277 units 6

NOT FOR SALE

Chapter 1

Functions ctions and Their Graphs

27. d1

1 3 2

3 2

3 2 2

2 4

1 2 2

3 4

4 25

25 4 2

32. (a)

29 29

9 49

28. d1

2 9 3

4 7 9

4 36

2 2

3 7

36 4

16 16

29. (a)

(b) d

1 6

12 0

Â§ 1 6 12 0 Â· (c) Â¨ , Â¸ 2 Â¹ Â© 2

25 144

Â§7 Â· Â¨ , 6Â¸ Â©2 Â¹

33. (a)

(5 ( 3)) 2 (6 6) 2

(b) d

Â§ 6 6 5 ( 3) Â· (c) Â¨ , Â¸ 2 Â© 2 Â¹ 30. (a)

(6, 1)

5 1

4 2

36 4

2 10

Â§ 1 5 2 4 Â· (c) Â¨ , Â¸ 2 Â¹ Â© 2

2, 3

(1, 4)

(8, 4)

34. (a)

4 2 x 2

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;4

(b) d

(4 4) 2 (8 1) 2

Â§1 8 4 4 Â· , (c) Â¨ Â¸ 2 Â¹ Â© 2

Â§9 Â· Â¨ , 4Â¸ Â©2 Â¹

(b) d

10 10 2

64 64

31. (a)

Â§ 2 10 10 2 Â· (c) Â¨ , Â¸ 2 Â¹ Â© 2

(b) d

1 7 1

Â§ 9 1 7 1Â· (c) Â¨ , Â¸ 2 Â¹ Â© 2

64 36

8 2

6, 6

5, 4

NOT FOR SALE Section ection 1.1

35. (a)

38. d

Rectangular Coordinates

18 50 12

242 382 2020 2 505 | 45

The pass is about 45 yards.

(b) d

16.8 5.6

12.3 4.9

501.76 54.76

39. midpoint

2006, 27,343.5

556.52

Â§ 16.8 5.6 12.3 4.9 Â· (c) Â¨ , Â¸ 2 2 Â© Â¹

5.6, 8.6

36. (a)

Â§ 2002 2010 19,564 35,123 Â· , Â¨ Â¸ 2 2 Â© Â¹

In 2006, the sales for the Coca-Cola Company were about $27,343.5 million. 40. midpoint

Â§ x1 x2 y1 y2 Â· , Â¨ Â¸ 2 Â¹ Â© 2 Â§ 2008 2010 1.89 2.83 Â· , Â¨ Â¸ 2 2 Â© Â¹

2009, 2.36

In 2009, the earnings per share for Big Lots, Inc. were about $2.36. 41. 2 2, 4 5

(b) d

5Â· 4Â· Â§1 Â§ Â¨ Â¸ Â¨1 Â¸ 2Â¹ 3Â¹ Â©2 Â© 9

1 9

82 3

Â§ 5 2 1 2 4 3 1 Â· (c) Â¨ , Â¸ 2 2 Â© Â¹ 37. d

0, 1

2 2, 3 5 4, 2

1 2, 1 5 1, 4

42. 3 6, 6 3

7Â· Â§ Â¨ 1, Â¸ 6Â¹ Â©

1202 1502

5 6, 3 3

3 6, 0 3

1 6, 3 3

3, 3

1, 0

3, 3

5, 0

43. 7 4, 2 8

36,900 30 41 | 192.09

The plane flies about 192 kilometers.

2 2 7

3, 6

4, 2 8 2, 10

4, 4 8 2, 4

4, 4 8 3, 4

44. 5 10, 8 6

3 10, 6 6

7 10, 6 6

5, 2

7, 0

3, 0

NOT FOR SALE

Chapter 1

Functions ctions and Their Graphs

45. (a) The minimum wage had the greatest increase from 2000 to 2010.

(b) Minimum wage in 1990: $3.80 Minimum wage in 1995: $4.25 Â§ 4.25 3.80 Âˇ Percent increase: Â¨ Â¸(100) | 11.8% 3.80 ÂŠ Âš Minimum wage in 1995: $4.25 Minimum wage in 2011: $7.25 Â§ 7.25 4.25 Âˇ Percent increase: Â¨ Â¸(100) | 70.6% 4.25 ÂŠ Âš So, the minimum wage increased 11.8% from 1990 to 1995 and 70.6% from 1995 to 2011. (c) Minimum wage Minimum wage Â§ Percent ÂˇÂ§ Minimum wage Âˇ = + Â¨ Â¸Â¨ Â¸ | $7.25 0.706($7.25) | $12.37 in 2016 in 2011 ÂŠ increase ÂšÂŠ in 2011 Âš So, the minimum wage will be about $12.37 in the year 2016. (d) Answer will vary. Sample answer: No, the prediction is too high because it is likely that the percent increase over a 4-year period (2011â&#x20AC;&#x201C;2016) will be less than the percent increase over a 16-year period (1995â&#x20AC;&#x201C;2011). 46. (a)

2 xm x1

So, x2 , y2

x1 x2 and ym 2

47. Because xm

2 xm

x1 x2

y1 y2 we have: 2

2 ym 2 ym y1

2 xm

y1 y2 y2

x1 , 2 ym y1 .

(b) The point 65, 83 represents an entrance exam score of 65. (c) No. There are many variables that will affect the final exam score. 48. (a)

x2 , y2

2 xm

x1 , 2 ym y1

2 Â&#x2DC; 4 1, 2 1 2

7, 0

(b)

x2 , y2

2 xm

x1 , 2 ym y1

2 Â&#x2DC; 2 5 , 2 Â&#x2DC; 4 11

9, 3

NOT FOR SALE Section ection 1.1

Rectangular Coordinates

Â§ x x2 y1 y2 Â· 49. The midpoint of the given line segment is Â¨ 1 , Â¸. 2 Â¹ Â© 2 x x2 y y2 Â· Â§ Â§ x x2 y1 y2 Â· x1 1 y1 1 Â¸ The midpoint between x1 , y1 and Â¨ 1 , 2 , 2 Â¸ is Â¨Â¨ Â¸ 2 Â¹ Â© 2 Â© 2 2 Â¹

Â§ 3x1 x2 3 y1 y2 Â· , Â¨ Â¸. 4 4 Â© Â¹

y y2 Â§ x1 x2 Â· Â§ x x2 y1 y2 Â· x2 1 y2 Â¸ The midpoint between Â¨ 1 , 2 2 Â¸ and x2 , y2 is Â¨Â¨ , Â¸ 2 Â¹ Â© 2 Â© 2 2 Â¹

Â§ x1 3 x2 y1 3 y2 Â· , Â¨ Â¸. 4 4 Â© Â¹

Â§ 3x x2 3 y1 y2 Â· Â§ x1 x2 y1 y2 Â· Â§ x1 3 x2 y1 3 y2 Â· , , , So, the three points are Â¨ 1 Â¸, Â¨ Â¸, and Â¨ Â¸. 4 4 2 Â¹ 4 4 Â© Â¹ Â© 2 Â© Â¹ Â§ 3x x2 3 y1 y2 Â· 50. (a) Â¨ 1 , Â¸ 4 4 Â© Â¹ Â§ x1 x2 y1 y2 Â· , Â¨ Â¸ 2 Â¹ Â© 2

Â§ 3 Â&#x2DC; 1 4 3 2 1 Â· , Â¨ Â¸ 4 4 Â© Â¹ Â§1 4 2 1Â· , Â¨ Â¸ 2 Â¹ Â© 2

Â§7 7Â· Â¨ , Â¸ Â© 4 4Â¹

Â§ 5 3Â· Â¨ , Â¸ Â© 2 2Â¹

Â§ x1 3 x2 y1 3 y2 Â· , Â¨ Â¸ 4 4 Â© Â¹

Â§ 1 3 Â&#x2DC; 4 2 3 1 Â· , Â¨ Â¸ 4 4 Â© Â¹

Â§ 13 5 Â· Â¨ , Â¸ 4Â¹ Â©4

Â§ 3x x2 3 y1 y2 Â· , (b) Â¨ 1 Â¸ 4 4 Â© Â¹

Â§ 3 2 0 3 3 0 Â· , Â¨ Â¸ 4 4 Â© Â¹

Â§ 3 9Â· Â¨ , Â¸ Â© 2 4Â¹

Â§ x1 x2 y1 y2 Â· , Â¨ Â¸ 2 Â¹ Â© 2

Â§ 2 0 3 0 Â· , Â¨ Â¸ 2 Â¹ Â© 2

Â§ x1 3 x2 y1 3 y2 Â· , Â¨ Â¸ 4 4 Â© Â¹

3Â· Â§ Â¨ 1, Â¸ 2Â¹ Â©

Â§ 2 0 3 0 Â· , Â¨ Â¸ 4 Â¹ Â© 4

Â§ 1 3Â· Â¨ , Â¸ Â© 2 4Â¹

(a) The point is reflected through the y-axis.

51.

(b) The point is reflected through the x-axis. (c) The point is reflected through the origin.

52. (a)

(b)

First Set d A, B

2 2

3 6

d B, C

2 6 2

6 3

d A, C

2 6

3 3

Because 32 42 of a right triangle.

16 9 16

52 , A, B, and C are the vertices First set: Not collinear

Second Set d A, B

8 5

3 2

d B, C

5 2 2

2 1

d A, C

8 2

3 1

10 10

Second set: The points are collinear. (c) If A, B, and C are collinear, then two of the distances will add up to the third distance.

A, B, and C are the vertices of an isosceles triangle 2 10 40. or are collinear: 10 10

NOT FOR SALE

Chapter 1

Functions ctions and Their Graphs

53. No. It depends on the magnitude of the quantities measured. 54. The y-coordinate of a point on the x-axis is 0. The x-coordinates of a point on the y-axis is 0. 55. False, you would have to use the Midpoint Formula 15 times. 56. True. Two sides of the triangle have lengths

the third side has a length of

149 and

58. (a) Because x0 , y0 lies in Quadrant II, x0 , y0 must

lie in Quadrant III. Matches (ii). (b) Because x0 , y0 lies in Quadrant II, 2 x0 , y0

must lie in Quadrant I. Matches (iii).

lie in Quadrant II. Matches (iv). (d) Because x0 , y0 lies in Quadrant II, x0 , y0

must lie in Quadrant IV. Matches (i).

18.

57. False. The polygon could be a rhombus. For example, consider the points 4, 0 , 0, 6 , 4, 0 , and 0, 6 .

59. Use the Midpoint Formula to prove the diagonals of the parallelogram bisect each other.

Â§b a c 0Âˇ , Â¨ Â¸ 2 Âš ÂŠ 2

Â§a b cÂˇ , Â¸ Â¨ 2Âš ÂŠ 2

Â§a b 0 c 0Âˇ , Â¨ Â¸ 2 2 Âš ÂŠ

Â§a b cÂˇ , Â¸ Â¨ 2Âš ÂŠ 2

Section 1.2 Graphs of Equations 1. solution or solution point

9. (a)

2, 0 : 2 2

3 2 2

2. graph

4 6 2

3. intercepts

(b)

2, 8 : 2 2

3 2 2

8 ?

12 z 8 ?

No, the point is not on the graph.

0 4

2 2

Yes, the point is on the graph.

5, 3 :

4 6 2

6. numerical

(b)

Yes, the point is on the graph.

5. circle; h, k ; r

0, 2 :

0 ?

4. y-axis

7. (a)

10. (a) 1, 5 : 5

5 4

8. (a) 1, 2 : 2

5 1

4 1

No, the point is not on the graph.

Yes, the point is on the graph.

4 1 2

5 z 3

(b)

6, 0 :

0 ?

4 6 2

4 4

Yes, the point is on the graph.

Yes, the point is on the graph. (b)

5, 0 :

5 5

0 0

Yes, the point is on the graph.

NOT FOR SALE Section 1.2

11. (a)

2, 3 :

15. y

2 1 2

3 ?

2 x 5

1 2

5 2

(x, y)

1, 7

0, 5

1, 3

2, 1

52 , 0

Yes, the point is on the graph. (b)

Graphs of Equations

1, 0 :

1 1 2

0 ?

2 2

0 z 4 No, the point is not on the graph. ?

12. (a) 1, 2 : 2 1 2 3

3 z 0 No, the point is not on the graph. ?

(b) 1, 1 : 2 1 1 3

16. y

2 1 3 0

Yes, the point is on the graph. 13. (a)

3, 2 : 3 2

9 4

3 x 4

(x, y)

2, 52

4 3

â&#x20AC;&#x201C;1

1 2

0, 1

1, 14 43 , 0 2, 12

20 ?

13 z 20 No, the point is not on the graph. (b)

4, 2 : 4 2

16 4

20 ?

20 17. y

Yes, the point is on the graph. 14. (a)

2, 163 :

1 3

2 3

1 3

2 2

Â&#x2DC;8 2Â&#x2DC;4 8 3 8 3

16 3 16 3

8 24 3 16 3

â&#x20AC;&#x201C;2

(x, y)

1, 4 0, 0 1, 2 2, 2 3, 0

16 3

x 2 3x

16 3 16 3

Yes, the point is on the graph. (b)

3, 9 : 13 3 3 1 3

2 3

2 9

9 18

9 ?

27 z 9 No, the point is not on the graph.

NOT FOR SALE

Chapter 1

Functions ctions and Their Graphs

20. x-intercepts: r 2, 0

5 x2

18. y

â&#x20AC;&#x201C;1

x, y

2, 1

1, 4

0, 5

1, 4

2, 1

y-intercept: 0, 16

21. x-intercept: 2, 0

y-intercept: 0, 2

22. x-intercept: 4, 0

y-intercepts: 0, r 2

23. x-intercept: 1, 0

y-intercept: 0, 2

24. x-intercepts: 0, 0 , 0, r 2

y-intercept: 0, 0

19. x-intercept: 3, 0

y-intercept: 0, 9

25. x 2 y

x 2

0 Â&#x; x2 y

0 Â&#x; y -axis symmetry

x y

0 Â&#x; x y

0 Â&#x; No x-axis symmetry

26. x y 2

27. y y

0 Â&#x; x2 y

0 Â&#x; x y2

0 Â&#x; x y

x 3

x3 Â&#x; No x-axis symmetry

Â&#x; y

x3 Â&#x; y

x 3 Â&#x; y 2

x3 Â&#x; Origin symmetry

x4 x2 3 x4 x2 3 Â&#x; y

0 Â&#x; No origin symmetry

x3 Â&#x; No y -axis symmetry

29. y

0 Â&#x; x-axis symmetry

0 Â&#x; x y2

Â&#x; y

x3 Â&#x; y

0 Â&#x; No y -axis symmetry

28. y

0 Â&#x; No origin symmetry

x y

x x2 1 x

x 4 x 2 3 Â&#x; No x-axis symmetry

x 3 Â&#x; y

Â&#x; y

x 4 x 2 3 Â&#x; y -axis symmetry x 4 x 2 3 Â&#x; No origin symmetry

x Â&#x; No y -axis symmetry x2 1

x x Â&#x; y Â&#x; No x-axis symmetry x2 1 x2 1 x x x Â&#x; y Â&#x; y Â&#x; Origin symmetry 2 2 2 1 x x 1 x 1

INSTRUCTOR USE ONLY y

NOT FOR SALE Section 1.2

30. y

1 1 x2 1 1 x

y y

1 Â&#x; y -axis symmetry 1 x2

0 Â&#x; xy 2 10

x y 10 2

x y

32. xy

Â&#x; y

1 1 Â&#x; y Â&#x; No x-axis symmetry 1 x2 1 x2 1 1 Â&#x; y Â&#x; No origin symmetry 2 1 x2 1 x

31. xy 2 10

x y

Graphs of Equations

0 Â&#x; xy 2 10

0 Â&#x; No y -axis symmetry 0 Â&#x; x-axis symmetry

0 Â&#x; xy 10 2

0 Â&#x; No origin symmetry

36.

x y 4 x y

4 x y

Â&#x; xy

4 Â&#x; No y -axis symmetry

Â&#x; xy

4 Â&#x; No x -axis symmetry

4 Â&#x; xy

4 Â&#x; Origin symmetry

33.

37. y

3 x 1

x-intercept:

13 , 0

y-intercept: 0, 1

No symmetry 34.

35.

38. y

2x 3

x-intercept:

32 , 0

y-intercept: 0, 3

No symmetry

NOT FOR SALE

Chapter 1

Functions ctions and Their Graphs

x2 2x

39. y

42. y

x3 1

x-intercepts: 0, 0 , 2, 0

x-intercept: 1, 0

y-intercept: 0, 0

y-intercept: 0, 1

No symmetry

â&#x20AC;&#x201C;1

x 3

43. y

x-intercept: 3, 0

y-intercept: none 40. y

x 2x 2

No symmetry

x-intercepts: 2, 0 , 0, 0

y-intercept: 0, 0

No symmetry

41. y

44. y

x3 3

x-intercept:

3, 0

1 x

x-intercept: 1, 0

y-intercept: 0, 1

y-intercept: 0, 3

No symmetry

No symmetry x

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;1

â&#x20AC;&#x201C;5

NOT FOR SALE Section 1.2

x 6

45. y

48. x

x-intercept: 5, 0

y-intercept: 0, 6

y-intercepts: 0,

No symmetry

x-axis symmetry

â&#x20AC;&#x201C;2

49. y

y2 5

x-intercept: 6, 0

Graphs of Equations

5 , 0,

1x 2

1 x

46. y

x-intercepts: 1, 0 , 1, 0

y-intercept: 0, 1

Intercepts: 10, 0 , 0, 5

y-axis symmetry 50. y

2x 3

Intercepts: 0, 1 , 51. y

y2 1

47. x

32 , 0

x2 4 x 3

x-intercept: 1, 0

y-intercepts: 0, 1 , 0, 1

x-axis symmetry x

â&#x20AC;&#x201C;1

Â±1

Â±2

Intercepts: 3, 0 , 1, 0 , 0, 3

52. y

x2 x 2

Intercepts: 2, 0 , 1, 0 , 0, 2

Chapter 1

53. y

NOT FOR SALE

Functions ctions and Their Graphs

2x x 1

Intercept: 0, 0

58. y

Intercepts: 0, 0 , 6, 0

x 3

59. y 54. y

4 x2 1

Intercepts: 3, 0 , 0, 3

Intercept: 0, 4

55. y

60. y

2 x

x 2

Intercepts: r 2, 0 , 0, 2

Intercepts: 8, 0 , 0, 2

56. y

x 1

61. Center: 0, 0 ; Radius: 4

0 y 0

x2 y2

62. Center: 0, 0 ; Radius: 5

0 y 0

x2 y2

Intercepts: 1, 0 , 0, 1

57. y

x 6

63. Center: 2, 1 ; Radius: 4

2 y 1

42 16

64. Center: 7, 4 ; Radius: 7

x 7

Intercepts: 0, 0 , 6, 0

y 4

7 y 4

72 49

NOT FOR SALE Section 1.2

65. Center: 1, 2 ; Solution point: 0, 0

x 1

y 2

1 0 2

1 y 2

Center: 0, 0 , Radius: 6

r2 r2 Â&#x; 5

70. x 2 y 2

Graphs of Equations

66. Center: 3, 2 ; Solution point: 1, 1

3 1

42 3

2 1

3 y 2

71.

1 y 3

Center: 1, 3 , Radius: 3

67. Endpoints of a diameter: 0, 0 , 6, 8

Â§ 0 6 0 8Â· Center: Â¨ , Â¸ 2 Â¹ Â© 2

3, 4

3 y 4

3 0 4

r 2 Â&#x; 25

3 y 4

68. Endpoints of a diameter: 4, 1 , 4, 1

1 2 1 2 1 2 1 2

4 4 2 8

1 1

72. x 2 y 1

Center: 0, 1 , Radius: 1

64 4 Â§1Â· Â¨ Â¸ 2 17 Â© 2Â¹

Midpoint of diameter (center of circle):

Â§ 4 4 1 1 Â· , Â¨ Â¸ 2 Â¹ Â© 2

0, 0

0 y 0

x2 y2

69. x 2 y 2

73.

x 12

Center: 12 , 12 , 2

1 2

Radius:

9 4 3 2

Center: 0, 0 , Radius: 5

74.

Chapter 1

NOT FOR SALE

Functions ctions and Their Graphs

2 y 3

Center: 2, 3 , Radius:

77. (a)

16 9 4 3

(b) 2 x 2 y

1040 3

1040 3 520 3

y A

2x x

5203 x

(d) When x

500,000 40,000t , 0 d t d 8

86 23 yards, the area is a maximum

of 751119 square yards. (e) A regulation NFL playing field is 120 yards long and 53 13 yards wide. The actual area is 6400 square 76. y

yards.

8000 900t , 0 d t d 6

78. (a)

(b) P

360 meters so:

2x 2 y w

360 y 180 x

x 180 x

(c)

(d) x

90 and y

A square will give the maximum area of 8100 square meters. (e) Answers will vary. Sample answer: The dimensions of a Major League Soccer field can vary between 110 and 120 yards in length and between 70 and 80 yards in width. A field of length 115 yards and width 75 yards would have an area of 8625 square yards.

NOT FOR SALE Section 1.2

Graphs of Equations

79. (a)

Because the line is close to the points, the model fits the data well. (b) Graphically: The point 90, 75.4 represents a life expectancy of 75.4 years in 1990. 0.002t 2 0.5t 46.6

Algebraically: y

0.002 90 0.5 90 46.6 2

75.4 So, the life expectancy in 1990 was about 75.4 years. (c) Graphically: The point 94.6, 76.0 represents a life expectancy of 76 years during the year 1994. y

0.002t 2 0.5t 46.6

76.0

0.002t 2 0.5t 46.6

0.002t 2 0.5t 29.4

Algebraically:

Use the quadratic formula to solve. b r

b 2 4ac 2a

0.5 r

0.5 2 4 0.002 29.4

2 0.002

0.5 r 0.0148 0.004 125 r 30.4

So, t

94.6 or t

(d) When t

115:

155.4. Since 155.4 is not in the domain, the solution is t

94.6 , which is the year 1994.

0.002t 2 0.5t 46.6 0.002 115 0.5 115 46.6 2

77.65 The life expectancy using the model is 77.65 years, which is slightly less than the given projection of 78.9 years. (e) Answers will vary. Sample answer: No. Because the model is quadratic, the life expectancies begin to decrease after a certain point.

NOT FOR SALE

Chapter 1

80. (a)

Functions ctions and Their Graphs

100

414.8

103.7

25.9

11.5

6.5

4.1

2.9

2.1

1.6

1.3

1.0

(b)

When x

85.5, the resistance is about 1.4 ohms.

85.5,

10,370 (85.5) 2

1.4 ohms.

(d) As the diameter of the copper wire increases, the resistance decreases. ax 2 bx3

81. y

a x b x

(a) y

y y

a x b x

ax bx 2

x y

x2 y 2

To be symmetric with respect to the y-axis; a can be any non-zero real number, b must be zero.

ax bx

x y Origin symmetry:

To be symmetric with respect to the origin; a must be zero, b can be any non-zero real number.

y-axis symmetry:

ax 2 bx3

(b) y

x2 y 2

82. x-axis symmetry:

x2 y2

x y

So, the graph of the equation is symmetric with respect to the x-axis, the y-axis, and the origin.

Section 1.3 Linear Equations in Two Variables 1. linear 2. slope 3. point-slope

9. (a) m

2. 3

Because the slope is positive, the line rises.

Matches L2 . (b) m is undefined. The line is vertical. Matches L3. (c) m

2. The line falls. Matches L1.

4. parallel 5. perpendicular

10. (a) m

(b) m 6. rate or rate of change 7. linear extrapolation 8. general

0. The line is horizontal. Matches L2 .

34 . Because the slope is negative, the line

falls. Matches L1. (c) m 1. Because the slope is positive, the line rises. Matches L3 .

Section 1.3 11.

17. y

Linear Equations in Tw Two Variables T

12 x 4 12

Slope: m

y-intercept: 0, 4

12.

3 2

18. Slope: m

y -intercept: 0, 6

13. Two points on the line: 0, 0 and 4, 6

Slope

y2 y1 x2 x1

6 4

3 2

14. The line appears to go through 0, 7 and 7, 0 . Slope

15. y

y2 y1 x2 x1

5x 3

Slope: m

19. y 3

y-intercept: 0, 3

16. Slope: m

0 7 7 0

y -intercept: 0, 10

0 3, horizontal line

Slope: m

y-intercept: 0, 3

20. x 5

0 5

Slope: undefined (vertical line) No y-intercept

NOT FOR SALE

Chapter 1

21. 5 x 2

Functions ctions and Their Graphs

24. 2 x 3 y

0 2, 5

vertical line

Slope: undefined No y-intercept

2 x 9

23 x 3

Slope: m

y-intercept: 0, 3

22. 3 y 5

Slope: m

6y y Slope: m

0 9 6 0

26. m

8 0 0 12

9 6

3 2

y-intercept: 0, 53

23. 7 x 6 y

25. m

8 12

2 3

8 4

30 7 x 30 7 x 6

7 6

y -intercept: 0, 5

27. m

6 2

1 3

NOT FOR SALE Section 1.3

28. m

29. m

30. m

31. m

4 4 4 2

7 7

8 5

5 1 4 2

4 1

6 6

m is undefined.

32. m

0 3

33. m

Linear Equations in Tw Two Variables T 0 10

4 0

1.6 3.1 5.2 4.8

1 § 4· ¨ ¸ 3 © 3¹ 3 11 2 2

5 2

1.5 10

34. m

5 0

35. Point: 2, 1 , Slope: m

Because m

0, 1 , 3, 1 ,

0.15

1 7

0, y does not change. Three points are

and 1, 1 .

36. Point: 3, 2 , Slope: m

Because m

0, y does not change. Three other points

are 4, 2 , 0, 2 , and 5, 2 . 37. Point: 8, 1 , Slope is undefined.

Because m is undefined, x does not change. Three points are 8, 0 , 8, 2 , and 8, 3 . 38. Point: 1, 5 , Slope is undefined.

Because m is undefined, x does not change. Three other points are 1, 3 , 1, 1 , and 1, 7 .

Chapter 1

Functions ctions and Their Graphs

39. Point: 5, 4 , Slope: m

45. Point: 3, 6 ; m

y increases by 2 for every one

y 6

unit increase in x. Three additional points are 4, 6 ,

Because m

3, 8 ,

2, 1

2 x 3

2 x

and 2, 10 .

40. Point: 0, 9 , Slope: m

2, y decreases by 2 for every one unit

Because m

increase in x. Three other points are 2, 5 ,

1, 11 , and 3, 15 . 41. Point: 1, 6 , Slope: m

12 , y decreases by 1 unit for every two

Because m

unit increase in x. Three additional points are 1, 7 ,

3, 8 ,

and 13, 0 .

42. Point: 7, 2 , Slope: m 1, 2

Because m

46. Point: 0, 0 ; m

y 0 y

4 x 0

1 2

y increases by 1 unit for every two

unit increase in x. Three additional points are 9, 1 ,

11, 0 ,

and 13, 1 .

43. Point: 0, 2 ; m

y 2 y

3 47. Point: 4, 0 ; m

3 x 0

3x 2

y 0 y

44. Point: 0, 10 ; m

1 x 0

y 10

x x 10

13 x 4

13 x

4 3

y 10 y

48. Point: 8, 2 ; m

y 2

1 4

y 2

1 x 4

1x 4

1 4

NOT FOR SALE Section 1.3

49. Point: 2, 3 ; m

54. Point: 5.1, 1.8 ; m

1 2

y 1.8

1 x 2

2 1 x 1 2 1 x 2 2

y 3

Linear Equations in Two Tw T Variables

y 3 y

50. Point: 2, 5 ; m

y 5

3 4

4 y 20

3 x 4

y 1

7 2

51. Point: 6, 1 ; m is undefined.

Because the slope is undefined, the line is a vertical line. 6

5 1 x 5

5 5 3 x 5 1 5 3 x 2 5

56. 4, 3 , 4, 4

y 3 y 3 y 3 y 52. Point: 10, 4 ; m is undefined.

Because the slope is undefined, the line is a vertical line. x

5 x 27.3

55. 5, 1 , 5, 5

5 x 5.1

3x 14

3 4

3x 6

4 3 x 4

4 4 7 x 4

8 7 7 x 8 2 7 1 x 8 2

57. 8, 1 , 8, 7

8, the slope is Because both points have x undefined, and the line is vertical.

52 ; m

53. Point: 4,

5 2

0 x 4

5 2

Chapter 1

NOT FOR SALE

Functions ctions and Their Graphs

58. 1, 4 , 6, 4

y 4

4 4 x 1

6 1

y 4

0 x 1

y 4

62. 8, 0.6 , 2, 2.4

10 y 6

3 x 8

10 3 x 8

10 y 6

3 x 24

10 y

3 x 18

y 0.6

Â§ 1Â· Â§1 5Â· 59. Â¨ 2, Â¸, Â¨ , Â¸ Â© 2Â¹ Â© 2 4Â¹

5 1 4 2 x 2

1 2 2 1 1 x 2 2 2 1 3 x 2 2

1 y 2 y y

2 1 3 x 1

6 1 1 x 1

3 1 1 x 3 3 1 4 x 3 3

y 1 y 1 y 1 y

3 9 x 10 5

0.3x 1.8

Â§1 Â· 63. 2, 1 , Â¨ , 1Â¸ Â©3 Â¹ y 1 y 1 y

2Â· Â§ 60. 1, 1 , Â¨ 6, Â¸ 3Â¹ Â©

2.4 0.6 x 8

2 8

y 0.6

1 1

x 2

1 2 3 0 1

The line is horizontal. Â§7 Â· Â§7 Â· 64. Â¨ , 8Â¸, Â¨ , 1Â¸ Â©3 Â¹ Â©3 Â¹ 1 8

7 7 3 3 7 3

9 and is undefined. 0

The line is vertical.

61. 1, 0.6 , 2, 0.6

0.6 0.6 x 1

2 1 0.4 x 1 0.6

0.4 x 0.2

y 0.6

65. L1 : y m1 L2 : y m2

1x 3

1 3

1x 3

1 3

The lines are parallel. 66. L1 : y

m1 L2 : y

4x 1 4 4x 7

INSTRUCTOR USE ONLY m2

The lines are parallel.

NOT FOR SALE Section 1.3

67. L1 : y m1 L2 : y m2

1x 2 1 2

m1 L2 : y

12 x 1 12

9 1 5 0

(a)

y 1 2

(b) 1 2

5 1

1 2

6 3

8 4

0 6 6 3

Â§ 7Âˇ L2 : 0, 1 , Â¨ 5, Â¸ ÂŠ 3Âš 7 1 3 m2 5 0

Slope: m

1, 3, 2

y 2

1 x 3

y 2

x 3

x 1

The lines are neither parallel nor perpendicular. 71. L1 : 3, 6 , 6, 0

12 x 2

x 7

(a) m

5 3 5 1

1, 3, 2

(b) m

x 5

y 75. 3x 4 y

7 34 x

The lines are parallel.

1 x 3

y 2

2 3

2 x 2

12 x 2

74. x y

23 , 78 , m y

(b)

7 8

34 x 32

34 x

23 , 78 , m y

7 4

Slope: m (a)

3 2

2, 1 , m y 1

L2 : 1, 3 , 5, 5

4 3

2x 3

70. L1 : 2, 1 , 1, 5

2, 1 , m

The lines are perpendicular.

Slope: m

L2 : 0, 3 , 4, 1

1 3 4 0

16 3 4

3 4

5 4

69. L1 : 0, 1 , 5, 9

1Âˇ Â§ L2 : 3, 5 , Â¨ 1, Â¸ 3Âš ÂŠ 1 5

3 m2 1 3

73. 4 x 2 y

The lines are perpendicular.

6 8

The lines are perpendicular.

54 5x 4

2 8 4 4

54 x

72. L1 : 4, 8 , 4, 2

The lines are neither parallel nor perpendicular. 68. L1 : y

Linear Equations in Two Tw T Variables

7 8

4 3

3 8

4 3

2 3

INSTRUCTOR USE ONLY y

4x 3

127 72

NOT FOR SALE

Chapter 1

76. 5 x 3 y

Functions ctions and Their Graphs

79. x y

5 x

53 x

Slope: m

(a)

Slope: m

53 ,

(a) m

24 y 18 24 y 18

40 x 35

24 y

40 x 53

40 y 30 40 y 30

24 x 21

40 y y

77. y 3

Slope: m (a)

24 x 9

9 40

1, 0 , m

0 4

is undefined.

81.

3, 2 , m

3.9, 1.4 , m

3 x 3.9

y 1.4

3 x 11.7

3x 13.1

3.9, 1.4 , m 1 3

1 3

x 3.9

y 1.4

1x 3

1.3

1x 3

0.1

x y 2 3 3x 2 y 6

1 0

x y 4 3 x y 12 12

3 4 4 x 3 y 12

is undefined.

3, 2 , m y

9 2

82. 3, 0 , 0, 4

Slope: m is undefined.

(b)

6 x 9

78. x 4

(a)

y 1.4

2.5

x 9.3

y 1.4

(b)

1, 0 , m

Slope: m

80. 6 x 2 y

1 x

3 5

3 x 5

x 4.3

2.5, 6.8 , m

x 78

24 x 78

3 4

1 x 2.5

y 6.8

(b)

53 24

(b)

3 7 3 , , 5 8 4

(b) m

2.5, 6.8 , m y

7 8

53 x

y 6.8

78 , 34

3 4

x 4

40 x 78

(a)

0 83.

x y 1 6 2 3 3 y 2 12 x 3 y 2 6x

12 Â&#x2DC; 1 0

1 1

NOT FOR SALE Section 1.3

Â§2 Âˇ 84. Â¨ , 0 Â¸, 0, 2

ÂŠ3 Âš

x y 2 3 2

x y c c x y

87. (a) m 135. The sales are increasing 135 units per year.

(b) m

0. There is no change in sales during the year.

40. The sales are decreasing 40 units per

year.

3x y 2 2 3x y 2 85.

Linear Equations in Two Tw T Variables

1 88. (a) greatest increase = largest slope

9, 36.54 , 10, 65.23

1, c z 0

65.23 36.54 10 9

28.69

1 2

So, the sales increased the greatest between the years 2009 and 2010.

least increase = smallest slope

x y

x y 3

8, 32.48 , 9, 36.54

86. d , 0 , 0, d , 3, 4

x y d d x y

36.54 32.48 9 8

4.06

So, the sales increased the least between the years 2008 and 2009.

1 (b)

4, 8.28 , 10, 65.23

65.23 8.28 10 4

56.95 | 9.49 6

3 4

x y

The slope of the line is about 9.49.

x y 1

6 x 100 6 100

200

12 feet

90. (a) and (b)

300

600

900

1200

1500

1800

2100

â&#x20AC;&#x201C;25

â&#x20AC;&#x201C;50

â&#x20AC;&#x201C;75

â&#x20AC;&#x201C;100

â&#x20AC;&#x201C;125

â&#x20AC;&#x201C;150

â&#x20AC;&#x201C;175

NOT FOR SALE

Chapter 1

Functions ctions and Their Graphs

50 25

600 300

25 300

1 12

1 x 600

12 1 x 50 12 1 x 12

y 50

y 50 y

1 , for every change in the horizontal measurement of 12 feet, the vertical 12 measurement decreases by 1 foot.

(d) Because m

(e)

1 | 0.083 12

91. 10, 2540 , m

8.3% grade

97. Using the points 0, 32 and 100, 212 , where the first

125

V â&#x20AC;&#x201C; 2540

125 t 10

V 2540

125t 1250

coordinate represents a temperature in degrees Celsius and the second coordinate represents a temperature in degrees Fahrenheit, you have

125t 3790, 5 d t d 10 m

92. 10, 156 , m

4.50

V â&#x20AC;&#x201C; 156

4.50 t 10

V 156

4.50t 45

4.5t 111, 5 d t d 10

0.07 S 2500

coordinate represents the year t and the second coordinate represents the value V, you have 0 875 175 5 0 175t 875, 0 d t d 5.

first coordinate represents the year t and the second coordinate represents the value V, you have 2,000 24,000 10 0

22,000 10

2200.

Since the point 0, 24,000 is the

300 2

150.

Using the point-slope form with m point 1, 970 , you have y y1

m t t1

y 970

150 t 1

y 970

150t 150

150 and the

150t 820.

(b) The slope is m 150. The slope tells you the amount of increase in the weight of average male childâ&#x20AC;&#x2122;s brain each year. (c) Let t

150 2 820

300 820

1120

The average brain weight at age 2 is 1120 grams.

V -intercept, b = 24,000, the equation is V

1270 970 3 1

96. Using the points 0, 24,000 and 10, 2000 , where the

32, the

98. (a) Using the points 1, 970 and 3, 1270 , you have

95. Using the points 0, 875 and 5, 0 , where the first

9 . 5

9 C 32. 5

equation is F

The slope measures the cost to produce one laptop bag.

180 100

Since the point 0, 32 is the F- intercept, b

93. The C-intercept measures the fixed costs of manufacturing when zero bags are produced.

94. W

212 32 100 0

2200t 24,000, 0 d t d 10.

(d) Answers will vary. (e) Answers will vary. Sample Answer: No. The brain stops growing after reaching a certain age.

NOT FOR SALE Section 1.3

99. (a) Total Cost = cost for

cost

+ cost

102. False. The lines are not parallel.

maintainance operator C

9.5t 11.5t 42,000

21.0t 42,000 Rate per hour Â&#x2DC; Hours

(b) Revenue R

45t

R C

45t 21t 42,000

24t 42,000

(d) Let P

101. False. The slope with the greatest magnitude corresponds to the steepest line.

purchase

for

fuel and

Linear Equations in Two Tw T Variables

8, 2

and 1, 4 : m1

0, 4

and 7, 7 : m2

4 2 1 8

7 4

7 0

2 7 11 7

103. Find the slope of the line segments between the points A and B, and B and C.

0, and solve for t.

0 42,000 1750

24t 42,000 24t t

The equipment must be used 1750 hours to yield a profit of 0 dollars. 100. (a) 10 m

(b) y

2 15 2 x 2 10 2 x

7 5 3 1

mBC

3 7 5 3

2 4 4 2

1 2 2

Since the slopes are negative reciprocals, the line segments are perpendicular and therefore intersect to form a right angle. So, the triangle is a right triangle.

x 15 m

mAB

8 x 50

104. On a vertical line, all the points have the same x-value, y2 y1 , you would have so when you evaluate m x2 x1 a zero in the denominator, and division by zero is undefined.

(c)

105. No. The slope cannot be determined without knowing the scale on the y-axis. The slopes will be the same if the scale on the y-axis of (a) is 2 12 and the scale on the

(d) Because m 8, each 1-meter increase in x will increase y by 8 meters.

y-axis of (b) is 1. Then the slope of both is 54 . 106. d1

x1 y2 y1

1 0

1 m 1

m1 0

x1 y2 y1

1 0

1 m 2

m2 0

Using the Pythagorean Theorem:

distance between 1, m1 , and 1, m2

2Âˇ 2Âˇ Â§ Â§ Â¨ 1 m1 Â¸ Â¨ 1 m 2 Â¸ ÂŠ Âš ÂŠ Âš

Â§ Â¨ ÂŠ

1 m1 1 m 2

m 1 2

m 2 2

m2 m1 2 m 2 2 2m1m2

2 1 m2

1 1 2

2 m2 m1 Â¸Âˇ Âš

m 1

2m1m2

INSTRUCTOR USE ONLY m1

NOT FOR SALE

Chapter 1

Functions ctions and Their Graphs

110. (a) Matches graph (ii).

107. No, the slopes of two perpendicular lines have opposite signs. (Assume that neither line is vertical or horizontal.) 108. Because 4 !

5 2

The slope is â&#x20AC;&#x201C;20, which represents the decrease in the amount of the loan each week. The y-intercept is 0, 200 , which represents the original amount of the

, the steeper line is the one with a

slope of â&#x20AC;&#x201C; 4. The slope with the greatest magnitude corresponds to the steepest line. 109. The line y

loan. (b) Matches graph (iii). The slope is 2, which represents the increase in the hourly wage for each unit produced. The y-intercept is 0, 12.5 , which represents the hourly rate if the

4 x rises most quickly.

employee produces no units. (c) Matches graph (i). The slope is 0.32, which represents the increase in travel cost for each mile driven. The y-intercept is 0, 32 , which represents the fixed cost of $30 per The line y

day for meals. This amount does not depend on the number of miles driven.

4 x falls most quickly.

(d) Matches graph (iv). The slope is â&#x20AC;&#x201C;100, which represents the amount by which the computer depreciates each year. The yintercept is 0, 750 , which represents the original purchase price.

The greater the magnitude of the slope (the absolute value of the slope), the faster the line rises or falls. 111. Set the distance between 4, 1 and x, y equal to the distance between 2, 3 and x, y .

4 ÂŞÂŹ y 1 ÂşÂź 2

4 y 1

ÂŞÂŹ x 2 ÂşÂź y 3

x 2 8 x 16 y 2 2 y 1

2 y 3

x2 4 x 4 y2 6 y 9

8 x 2 y 17

4 x 6 y 13

12 x 8 y 4

4 3 x 2 y 1

3x 2 y 1

This line is the perpendicular bisector of the line segment connecting 4, 1 and 2, 3 . 112. Set the distance between 6, 5 and x, y equal to the distance between 1, 8 and x, y .

6 y 5

x 2 12 x 36 y 2 10 y 25 x 2 y 2 12 x 10 y 61 12 x 10 y 61

1 y 8

x 2 2 x 1 y 2 16 y 64 x 2 y 2 2 x 16 y 65 2 x 16 y 65

10 x 26 y 4

2 5 x 13 y 2

5 x 13 y 2

NOT FOR SALE

Section 1.4 1.

Functions

52 and x, y equal to the distance between 7, 1 and x, y .

113. Set the distance between 3,

y 52

3 y

ÂŞÂŹ x 7 ÂşÂź y 1

7 y 1

x2 6x 9 y 2 5 y

25 4

x 2 14 x 49 y 2 2 y 1

6 x 5 y

61 4

14 x 2 y 50

24 x 20 y 61

56 x 8 y 200

80 x 12 y 139

52 and 7, 1 .

This line is the perpendicular bisector of the line segment connecting 3,

114. Set the distance between 12 , 4 and x, y equal to the distance between

1 2

y 4

x 12

x2 x

1 4

y 4

y 2 8 y 16

x 72 y 54

x2 7 x

49 4

65 4

x2 y 2 7 x

x 8y

65 4

7 x

39 16

128 x 168 y 39

21 y 2

x2 y 2 x 8 y

5 y 2

72 , 54 and x, y .

5 y 2

25 16

221 16

Section 1.4 Functions 1. domain; range; function

10. (a) The element c in A is matched with two elements, 2 and 3 of B, so it is not a function.

2. independent; dependent 3. implied domain

(b) Each element of A is matched with exactly one element of B, so it does represent a function.

4. difference quotient

(c) This is not a function from A to B (it represents a function from B to A instead). (d) Each element of A is matched with exactly one element of B, so it does represent a function.

5. Yes, the relationship is a function. Each domain value is matched with exactly one range value. 6. No, the relationship is not a function. The domain value of â&#x20AC;&#x201C;1 is matched with two output values. 7. No, it does not represent a function. The input values of 10 and 7 are each matched with two output values.

11. x 2 y 2

4 Â&#x; y

r 4 x2

No, y is not a function of x. 12. x 2 y

4 Â&#x; y

4 x2

Yes, y is a function of x. 8. Yes, the table does represent a function. Each input value is matched with exactly one output value.

13. 2 x 3 y

1 3

2 x

Yes, y is a function of x.

9. (a) Each element of A is matched with exactly one element of B, so it does represent a function.

(b) The element 1 in A is matched with two elements, â&#x20AC;&#x201C;2 and 1 of B, so it does not represent a function.

4 Â&#x; y

14.

2 y 2

4 x 2

No, y is not a function of x.

INSTRUCTOR USE ONLY (d) The element 2 in A is not matched with an element B,, so the relation does not represent a function. of B

NOT FOR SALE

Chapter 1

Functions ctions and Their Graphs

25. f y

16 x 2

15. y

(a) f 4

Yes, y is a function of x. x 5

16. y

Yes, y is a function of x. 4 x

17. y

4 x Â&#x; y

4 x

Yes, y is a function of x. 20. x 1

0.25

4 x2

(a) f 8

(a) f 1

2 1 3

(b) f 3

2 3 3

22. V r

4 S 3

4S 3 3 2

23. g t

2x 5

4 S 3

36S

9 S 2

4 S 27 3 8

4 S 3

8r 3

1 0 9

(b) q 3

1 is undefined. 32 9

1 9

y 3 9 2

1 y2 6 y

2 2 3 2

8 3 4

11 4

2 0 3 2

0 2

Division by zero is undefined. 2 x 3 2

2x2 3 x2

x 2

4 2 3 2 5 2

(b) g t 2

29. f x

4 t 2 3 t 2 5 2

4t 2 3t 5 15 4t 3t 10

2 2

22 2 2

1.5

2 1.5

2 2 x 2

30. f x

0.75 x 2x

Â 1, if x 1 Â® if x ! 1 Â¯1,

x 1 x 1

x 4

(a) f 2

t 2 2t

(b) f 2

(b) h 1.5

(a) f 2

4t 2 19t 27

(a) h 2

x 2

2t 2 3 t2

(b) q 0

24. h t

8 8 2

(a) q 2

4t 2 3t 5

(a) g 2

8 2

(a) q 0

28. q t

32 S r3 3

8 2

2 x 1 3

4S r 3 3

(a) V 3

(b) V

3 2 x

1 x2 9

27. q x

2x 3

2.5

x 8 2

No, this is not a function of x. 21. f x

75 0 x

(b) f 0.25

(b) f 1

No, y is not a function of x. 19. y

y 3

26. f x

Yes, y is a function of x. 18.

2 4

(b) f 2

2 4

x2 4

NOT FOR SALE

Section 1.4 1.

31. f x

35. f x

Â2 x 1, x 0 Â® Â¯2 x 2, x t 0

(a) f 1

2 1 1

f 2

12 2 4

12 1 4

4 12

9 2

2 0 2

2 2 2

f 1

f 0

12 0 4

f 1

1 2

f 2

32. f x

(a) f 3

4 5 3

(b) f 4

4 2

33. f x

f x

f 1

1 2

f 0

f 1

1 2

f 2

2 3

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;3

â&#x20AC;&#x201C;2

3 3

f 1

9 1

f 2

9 2

f 3

3 3 4 3 5 3

1 2

f 5

f x

t 3

h 5

1 2

5 3

h 4

1 2

4 3

1 2

h 3

1 2

3 3

h 2

1 2

2 3

1 2

h 1

1 2

1 3

t h t

â&#x20AC;&#x201C;5 1

â&#x20AC;&#x201C;4 1 2

â&#x20AC;&#x201C;1

9 2

8 2

5 0 1 2

37. 15 3 x 34. h t

ÂÂ°9 x 2 , x 3 Â® Â°Ì&#x201E;x 3, x t 3

x f x

f 4

4 1

â&#x20AC;&#x201C;2

36. f x

f 2

x2 3

1 Â°Â 2 x 4, x d 0 Â® 2 Â°Ì&#x201E; x 2 , x ! 0

(b) f 0

Â4 5 x, x d 2 Â° 2 x d 2 Â®0, Â°x 2 1, x ! 2 Â¯

Functions

â&#x20AC;&#x201C;3 0

38.

f x

5x 1

0 15

x 39.

â&#x20AC;&#x201C;2 1 2

â&#x20AC;&#x201C;1 1

3x 4 5 3x 4 x

40.

f x

12 x 2 5 x2 x

0 0 4 3 12 x 2 5 0 12 r 12

r2 3

NOT FOR SALE

Chapter 1

41. x 2 9

Functions ctions and Their Graphs

2 x 1

x 2 x

7x 5

x 5x 6

3 x 2

x x 1

x 3

x x 1 x 1

1, or x

f x

44.

x 2 x 1 4 x 1

1 x 2 4

0 Â&#x; x

x2 4

0 Â&#x; x

f x

x 3

49. f x

f x

2x2

x4 4x2

x x 4

x x 2 x 2

x2 x 2 x 2

2 x

x 6

0 Â&#x;

3, which is a contradiction, since

x 2

0 Â&#x;

2 Â&#x; x

53. g y

y 10

Domain: y 10 t 0 y t 10 The domain is all real numbers y such that y t 10.

Because f x is a polynomial, the domain is all real numbers x.

54. f t

t 4

Because f t is a cube root, the domain is all real

4 t

numbers t.

The domain is all real numbers t except t s y

x represents the principal square root.

numbers x.

52.

Because f x is a polynomial, the domain is all real

51. h t

0 Â&#x; x 0 Â&#x; x 0 Â&#x; x

x 3

1 2x

g x

x 4 x 2

g x

x4 2x2 2

5x2 2 x 1

50. f x

x 1

0 x 2

47.

x3 x 2 4 x 4

48.

g x

x 2x 1

f x

0, x

x3 x

46.

0 Â&#x; x

43.

0 x 1

0 Â&#x; x

x 3

x x 2

5 x 3

x 5

x 2 8 x 15

x 2

f x

42.

g x

f x

45.

3y y 5

55. g x

1 3 x x 2

The domain is all real numbers x except x

0, x

y 5 z 0 y z 5

INSTRUCTOR USE ONLY The domain is all real numbers y except y

NOT FOR SALE

Section 1.4 1.

10 x 2x

h x

56.

x2 2 x z 0

Domain: x 6 t 0 Â&#x; x t 6 and x z 6

x x 2 z 0

The domain is all real numbers x except x 57. f s

x 6 6 x

58. f x

Functions

0, x

s 1 s 4

The domain is all real numbers x such that x ! 6 or 6, f . x 4 x

59. f x

Domain: s 1 t 0 Â&#x; s t 1 and s z 4 The domain consists of all real numbers s, such that s t 1 and s z 4.

The domain is all real numbers x such that x ! 0 or 0, f . x 2 x 10

60. f x

x 10 ! 0 x ! 10

The domain is all real numbers x such that x ! 10. 61. (a)

Height, x

Volume, V

484

800

972

1024

980

864

The volume is maximum when x

4 and V

1024 cubic centimeters.

(b)

V is a function of x. (c) V

x 24 2 x

Domain: 0 x 12

NOT FOR SALE

Chapter 1

Functions ctions and Their Graphs

66. (a) V l Â&#x2DC;wÂ&#x2DC;h xÂ&#x2DC; yÂ&#x2DC; x x 2 y where 4 x y 108. So, y 108 4 x and

62. (a) The maximum profit is $3375.

(b)

x 2 108 4 x

108 x 2 4 x3.

Domain: 0 x 27 (b)

Yes, P is a function of x. Revenue Cost

Â§ price ÂˇÂ§ number Âˇ Â§ number Âˇ Â¨ Â¸Â¨ Â¸ cost Â¨ Â¸ per unit of units ÂŠ ÂšÂŠ Âš ÂŠ of units Âš

(c) The dimensions that will maximize the volume of the package are 18 u 18 u 36. From the graph, the maximum volume occurs when x 18. To find the dimension for y, use the equation y 108 4 x.

ÂŞÂŹ90 x 100 0.15 ÂşÂź x 60 x, x ! 100

90 0.15 x 15 x 60 x 105 0.15 x x 60 x

Â§PÂˇ Â¨ Â¸ ÂŠ4Âš

1 bh 2

67. A

65.

Â§C Âˇ SÂ¨ Â¸ ÂŠ 2S Âš

y y 30

1 xy 2

line, the slopes between any pair are equal.

4s Â&#x;

P 4

1 y 2 0 1 y 2

P2 16 2S r

C 2S

Because 0, y , 2, 1 , and x, 0 all lie on the same

S r2, C

108 72

45 x 0.15 x 2 , x ! 100

64. A

108 4 18

105 x 0.15 x 2 60 x

s 2 and P

63. A

108 4 x

1 x2 10

So, A

C2 4S

1 Â§ x Âˇ xÂ¨ Â¸ 2 ÂŠ x 2Âš

x2 . 2 x 2

The domain of A includes x-values such that x 2 ÂŹÂŞ2 x 2 ÂźÂş ! 0. By solving this inequality, the

3x 6

1 30 10 3 30 6

0 1 x 2 1 x 2 2 1 x 2 x x 2

6 feet

If the child holds a glove at a height of 5 feet, then the ball will be over the child's head because it will be at a height of 6 feet.

domain is x ! 2. 68. A

l Â&#x2DC;w

2 x y

2 xy

But y 36 x 2 , so A domain is 0 x 6.

2 x 36 x 2 . The

NOT FOR SALE

Section 1.4 1.

Functions

69. For 2004 through 2007, use p t

4.57t 27.3.

2004: p 4

4.57 4 27.3

45.58%

2005: p 5

4.57 5 27.3

50.15%

2006: p 6

4.57 6 27.3

54.72%

2007: p 7

4.57 7 27.3

59.29%

For 2008 through 2010, use p t

3.35t 37.6.

2008: p 8

3.35 8 37.6

64.4%

2009: p 9

3.35 9 37.6

67.75%

2010: p 10

3.35 10 37.6

71.1%

70. For 2000 through 2006, use

p t

0.438t 2 10.81t 145.9.

2000: p 0

0.438 0 10.81 0 145.9

$145.9 thousand

2001: p 1

0.438 1 10.81 1 145.9

$157.148 thousand

2002: p 2

0.438 2 10.81 2 145.9

$169.272 thousand

2003: p 3

0.438 3 10.81 3 145.9

$182.272 thousand

2004: p 4

0.438 4 10.81 4 145.9

$196.148 thousand

2005: p 5

0.438 5 10.81 5 145.9

$210.9 thousand

2006: p 6

0.438 6 10.81 6 145.9

$226.528 thousand

For 2007 though 2010, use p t

5.575t 2 110.67t 720.8.

2007: p 7

5.575 7 110.67 7 720.8

$219.285 thousand

2008: p 8

5.575 8 110.67 8 720.8

$192.24 thousand

2009: p 9

5.575 9 110.67 9 720.8

$176.345 thousand

2010: p 10

71. (a) Cost

5.575 10 110.67 10 720.8 2

variable costs fixed costs 12.30 x 98,000

(b) Revenue R

price per unit u number of units

$171.6 thousand

72. (a) Model: Total cost

Fixed costs

Labels:

17.98 x Revenue Cost

Total cost

Fixed cost

6000

Variable costs

17.98 x 12.30 x 98,000

5.68 x 98,000

Equation: (b) C

C x

Variable costs

0.95 x

6000 0.95 x

6000 0.95 x x

6000 0.95 x

Chapter 1

NOT FOR SALE

Functions ctions and Their Graphs

73. (a)

(b)

3000

d2 d 2 3000

Domain: d t 3000 (because both d t 0 and d 2 3000 t 0) 2

74. F y

(a)

149.76 10 y 5 2

F y

26,474.08

149,760.00

847,170.49

2,334,527.36

4,792,320

The force, in tons, of the water against the dam increases with the depth of the water. (b) It appears that approximately 21 feet of water would produce 1,000,000 tons of force. 149.76 10 y 5 2

1,000,000

(c)

1,000,000 149.76 10

y5 2

2111.56 | y 5 2 21.37 feet | y n rate

75. (a) R

12.00n 0.05n 2

(b)

n ÂŞÂŹ8.00 0.05 n 80 ÂşÂź , n t 80 12n

240n n 2 , n t 80 20

n2 20

100

110

120

130

140

150

R n

$675

$700

$715

$720

$715

$700

$675

The revenue is maximum when 120 people take the trip. 76. (a)

f 2010 f 2003

2010 2003

98.7 52.9 7 45.8 7 | 6.54

6.54t 33.3

Approximately 6.54 million more tax returns were made through e-file each year from 2003 to 2010. (b)

52.9

59.5

66.0

72.5

79.1

85.6

92.2

98.7

(e) The algebraic model is a good fit to the actual data. (f) y

6.65 x 34.2; The models are similar.

NOT FOR SALE

Section 1.4 1.

f x

77.

f 2 h

x2 x 1

2 h 2

f x

78.

2 h 1

f 5 h

5 5 h 5 h

25 5h 25 10h h 2

h 3h 3

25 5h 25 10h h 2

2 1

f 2 h f 2

h 2 3h

f 2 h f 2

h 2 3h h

h 2 5h

3 f 5

5 5 5

f 5 h f 5

h 5h h h h 5

f x h

f x

25 25

h 3, h z 0

79.

4 4h h 2 2 h 1 2

5x x2

f 2

Functions

h 5 , h z 0

x3 3x

h 3 x h

x3 3x 2 h 3 xh 2 h3 3x 3h f x h f x

x3 3x 2h 3xh2

h3 3 x 3h x3 3x

h 3 x 3 xh h 3

h 3 x 3 xh h 2 3, h z 0 2

f x

80.

f x h

4x2 2x 4 x h 2 x h

4 x 2 2 xh h 2 2 x 2h 4 x 2 8 xh 4h 2 2 x 2h f x h f x

4 x 2 8 xh 4h 2 2 x 2h 4 x 2 2 x h 8 xh 4h 2 2h h h 8 x 4h 2

h 8 x 4h 2, h z 0

81.

g x

g x g 3

x 3

1 x2 1 1 9 x2 x 3 9 x2 9 x 2 x 3

x 3 x 3

82.

f t

f 1

f t f 1

t 1

1 t 2 1 1 1 2 1 1

t 2 t 1 1 t 2

9 x 2 x 3

x 3 ,x z 3 9x2

2 t 1

1 ,t z 1 t 2

NOT FOR SALE

Chapter 1

83. f x

Functions ctions and Their Graphs

f x f 5

5x 5 ,x z 5 x 5

x 5

84.

f x

x2 3 1

f 8

82 3 1

f x f 8

x 8

c 4 Â&#x; c

have 32

x2 3 4 ,x z 8 x 8

2. So, g x

2 x 2 .

14 are on

cx. Because 0, 0 and 1,

the line, the slope is 14 . So, f x

1 x. 4

32. So, r x

c 4 Â&#x; c

x . Because

2 and

and the corresponding y-values are 6 and 3, c h x

x and y. However, y a function.

radicand can be any real number. So, the domain of g is all real numbers x and the domain of f is all real numbers x such that x t 2. 95. No; x is the independent variable, f is the name of the function.

0.5, h | 20 feet and

1.25, h | 28 feet.

3 and

x 2 4 is a relation between r

cannot be negative. g x is an odd root, therefore the

when t

89. False. The equation y 2

94. Because f x is a function of an even root, the radicand

(b) Using the graph when t

32 x.

88. By plotting the data, you can see that they represent

h x

Domain: x ! 1

96. (a) The height h is function of t because for each value of t there is a corresponding value of h for 0 d t d 2.6.

87. Because the function is undefined at 0, we have r x

c x. Because 4, 8 is on the graph, you

have 8

g x

a zero to occur in the denominator which results in the function being undefined.

86. By plotting the data, you can see that they represent a

line, or f x

Domain: x t 1

1 x 1

as it is possible to find the square root of 0. However, 1 cannot be included in the domain of g x as it causes

85. By plotting the points, we have a parabola, so g x

cx 2 . Because 4, 32 is on the graph, you 2

x 1

The value 1 may be included in the domain of f x

x2 3 1 5 x 8

93. f x

x 2 4 does not represent

90. True. A function is a relation by definition. 91. False. The range is > 1, f . 92. True. The set represents a function. Each x-value is mapped to exactly one y-value.

(d) No, the time t is not a function of the height h because some values of h correspond to more than one value of t. 97. (a) Yes. The amount that you pay in sales tax will increase as the price of the item purchased increases. (b) No. The length of time that you study the night before an exam does not necessarily determine your score on the exam. 98. (a) No. During the course of a year, for example, your salary may remain constant while your savings account balance may vary. That is, there may be two or more outputs (savings account balances) for one input (salary). (b) Yes. The greater the height from which the ball is dropped, the greater the speed with which the ball will strike the ground.

Section 1.5 Analyzing Graphs of Functions 1. Vertical Line Test

7. Domain: f, f ; Range: > 4, f

2. zeros

(a) f 2

3. decreasing

(b) f 1

4. maximum

5. average rate of change; secant

(d) f 1

0 2

6. odd

NOT FOR SALE Section 1.5

8. Domain: f, f ; Range: f, f

(a) f 1

(d) f 1

(b) f 1

2 3

(a) f 2

(d) f 2

3x 2

0 Â&#x; x

2 3

x 8

0 Â&#x; x

x 9x2 4

f x

0 0

x 2

0 Â&#x; x

f x

0 2 0

x x 2 2

x2 2

0 or

x 1

13. x y

A vertical line intersects the graph more than once, so y is not a function of x. 14. x 2

2 x 2 7 x 30

2 x

5 x 6

2x 5 x

0 52

2 x 2 7 x 30

x 3 4 x 2 9 x 36

x x 4 9 x 4

4 x 2 9

x x 4

A vertical line intersects the graph at most once, so y is a function of x. f x

x3 4 x 2 9 x 36

2 xy 1

15.

f x

20. 2

r 2

y is not a function of x. Some vertical lines intersect the graph twice. 2

1 x3 2

x 2x

0 Â&#x; x

A vertical line intersects the graph at most once, so y is a function of x.

x 7

1 x3 2

1Â&#x; y

x 2 9 x 14 4x

f x

19.

1 x3 4

12. x y

2 x 8

x 2 9 x 14 4x x 7 x 2

(b) f 1

3 x

x 9x2 4 x

10. Domain: f, f ; Range: â&#x20AC;&#x201C; f, 1@

11. y

18.

(d) f 1

x 9 2

0 Â&#x; x

r3 f x

21.

4 x3 24 x 2 x 6

4 x 3 24 x 2 x 6

4 x x 6 1 x 6

x 6

x 6 4 x x 6 2 x 1 2 x

0 or 2 x 1

x 6 x

3 x 2 22 x 16

3x 2 22 x 16

17.

9. Domain: f, f ; Range: 2, f

(a) f 2

f x

16.

(b) f 2

Analyzing Graphs of Functions

or 2 x 1

1 2

NOT FOR SALE

Chapter 1

Functions ctions and Their Graphs

f x

22.

9 x 4 25 x 2

x 9 x 25

0 Â&#x; x

9 x 25

0 Â&#x; x

r 53

f x

23.

2x 1

(b)

f x

2 x 11

1 2

3x 2

28. (a)

Zero: x

5 3

f x

5 x

3x 5

3 x 14 8

3 x 14

5 x

Zero: x

26. (a)

(b)

f x

3x 1 x 6 3x 1

Zeros: x f x

0, x

x 7

x x 7

3x 14 8

29. (a)

26 f x

(b)

25. (a)

11 2

3x 2

(b)

2 x 11

f x

(b)

11 2

Zero: x

2x 1

24.

27. (a)

0 Â&#x; x

1 3

3x 1 x 6 0 0 1 3

NOT FOR SALE Section 1.5

30. (a)

Analyzing Graphs of Functions

Â2 x 1, x d 1 Â® 2 Â¯x 2, x ! 1

38. f x

The function is decreasing on 1, 0 and increasing on

f, 1

Zeros: x

r2.1213

f x

2 x2 9 3 x

(b)

2 x2 9 3 x

2 x2 9

0 Â&#x; x

31. f x

39. f x

3 2 2

r2.1213 Constant on f, f

3 x 2

(b)

x2 4 x

The function is decreasing on f, 2 and increasing on

x f x

40. g x

â&#x20AC;&#x201C;2

(a)

2, f . 33. f x

(a)

The function is increasing on f, f . 32. f x

and 0, f .

x3 3x 2 2

The function is increasing on f, 0 and 2, f and decreasing on 0, 2 . 34. f x

Increasing on f, f

(b)

x2 1

The function is decreasing on f, 1 and increasing

â&#x20AC;&#x201C;2

g x

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;1

on 1, f . 35. f x

s2 4

41. g s

x 1 x 1

The function is increasing on 1, f .

(a)

The function is constant on 1, 1 . The function is decreasing on f, 1 . 36. The function is decreasing on 2, 1 and 1, 0 and

Decreasing on f, 0 ; Increasing on 0, f

increasing on f, 2 and 0, f . (b) 37. f x

Âx 3, x d 0 Â° 0 x d 2 Â®3, Â°2 x 1, x ! 2 Â¯

â&#x20AC;&#x201C;4

g s

The function is increasing on f, 0 and 2, f . The function is constant on 0, 2 .

Chapter 1

42. f x

Functions ctions and Their Graphs 46. f x

3x 4 6 x 2

x2 3

(a)

Increasing on 1, 0 , 1, f ; Decreasing on

f, 1 , 0, 1

(b)

x f x

Decreasing on â&#x20AC;&#x201C; f, 0 ; Increasing on 0, f

(b)

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;3

f x

47. f x

43. f x

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;1

1.59

3x 2 2 x 5

1 x

(a)

Relative minimum: Decreasing on f, 1

(b)

â&#x20AC;&#x201C;3

f x

44. f x

48. f x

â&#x20AC;&#x201C;1 3

13 , 163

or 0.33, 5.33

x 2 3x 2

x 3

(a)

Relative maximum: 1.5, 0.25

49. f x

2 x 2 9 x

Increasing on 2, f ; Decreasing on 3, 2

(b)

x f x

â&#x20AC;&#x201C;3

â&#x20AC;&#x201C;1

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;1.414

2 Relative maximum: 2.25, 10.125

45. f x

50. f x

x x 2 x 3

(a)

Increasing on 0, f

(b)

Relative minimum: 1.12, â&#x20AC;&#x201C; 4.06

Relative maximum: 1.79, 8.21

INSTRUCTOR NST TRUC USE ONLY f x

2.8

5.2

NOT FOR SALE Section 1.5

51. f x

x3 3x 2 x 1

Analyzing Graphs of Functions

56. f x

4x 2

f x t 0 on ÂŞÂŹ 12 , f 4x 2 t 0

4 x t 2 x t 12

Relative maximum: 0.15, 1.08

Relative minimum: 2.15, 5.08

52. h x

x3 6 x 2 15

ÂŞ 1 ÂŹ 2 , f 57. f x

9 x2

f x t 0 on > 3, 3@

Relative minimum: 4, 17

Relative maximum: 0, 15

53. h x

58. f x

x2 4x

f x t 0 on f, 0@ and >4, f

x2 4 x t 0 x x 4 t 0

Relative minimum: 0.33, 0.38

54. g x

f, 0@, >4, f

x 4 x

59. f x

Relative maximum: 2.67, 3.08

55. f x

4 x

f x t 0 on f, 4@

x 1

f x t 0 on >1, f

x 1 t 0 x 1 t 0 x t1

>1, f

60. f x

1 x

f x is never greater than 0. f x 0 for all x.

Chapter 1

61. f x

NOT FOR SALE

Functions ctions and Their Graphs

2 x 15

f 3 f 0

3 0

67. s0

9 15 3

(b)

5 1

23 7 4

16 4

3 is

The average rate of change from x1

5 is 4.

3 3

3 1

s 3 s 0

3 0

54 6 3

6, m

Secant line: y 6 0

The average rate of change from x1

1 to x2

3 is 0.

1 to x2

6 is 0.

(d) The slope of the secant line is positive. (e) s 0

x3 3x 2 x

f 3 f 1

64. f x

0 to x2

x2 2x 8

f 5 f 1

63. f x

16t 2 64t 6

(a) s

The average rate of change from x1 2. 62. f x

6, v0

16 t 0

16t 6

(f )

x3 6 x 2 x

f 6 f 1

6 6 0 0 6 1 5 5 The average rate of change from x1

65. (a)

68. (a) s

16t 2 72t 6.5

(b)

(b) To find the average rate of change of the amount the U.S. Department of Energy spent for research and development from 2005 to 2010, find the average rate of change from 5, f 5

to 10, f 10

. f 10 f 5

10 5

10,925 8501.25 5

484.75

The amount the U.S. Department of Energy spent for research and development increased by about $484.75 million each year from 2005 to 2010. 66. Average rate of change

s t2 s t1

4 0 second

38.5 6.5 4

32 4

0 to t

8 feet per

(d) The slope of the secant line through 0, s 0

and

4, s 4

is positive.

(e) The equation of the secant line: m

8, y

8t 6.5

(f )

t2 t1 s 9 s 0

9 0 540 0 9 0 60 feet per second.

As the time traveled increases, the distance increases rapidly, causing the average speed to increase with each time increment. From t 0 to t 4, the average speed is less than from t 4 to t 9. Therefore, the overall average from t 0 to t 9 falls below the average found in part (b).

NOT FOR SALE Section 1.5

69. v0

120, s0

71.

f x

16t 2 120t

(a) s

f x

(b)

Analyzing Graphs of Functions

x6 2 x 2 3

2 x 3

x6 2 x 2 3 f x

The function is even. y-axis symmetry. 72.

200 216 2

5 3 second

16 2

3 to t

g x

8 feet per

(f )

The function is odd. Origin symmetry. 73.

h x

5, 200 we have

x 5

x 5 x z h x

z h x

The function is neither odd nor even. No symmetry.

270

74.

f x

x 1 x2 x 1 x

x 1 x2

70. (a) s

x 5

8t 240.

5 x

x 5x

8 t 5

y 200

g x

5, s 5

is negative. Using 5, s 5

x3 5 x

(d) The slope of the secant line through 3, s 3

and

(e) The equation of the secant line: m

g x

f x

16t 80 2

(b)

The function is odd. Origin symmetry. 75. f s

4s3 2 4 s

z f s

2 1 per second

16 64 1

48 1

1 to t

48 feet

2, s 2

is negative. Using 1, s 1

1, 64

g s

4s 2 3

g s

4 s

4s 2 3

(e) The equation of the secant line: m

The function is neither odd nor even. No symmetry. 76.

(d) The slope of the secant line through 1, s 1

and

y 64

z f s

we have

g s

The function is even. y-axis symmetry.

48 t 1

48t 112.

(f )

Chapter 1

NOT FOR SALE

Functions ctions and Their Graphs

80. h x

77.

The graph of f x

9 is symmetric to the y-axis,

which implies f x is even. f x

The graph displays y-axis symmetry, which implies h x is even.

h x

f x

81. f x

5 3x

The graph displays no symmetry, which implies f x is neither odd nor even. f x

x 2

h x

1 x

f x

5 3 x

1 x

5 3x

1 x z f x

z f x

The function is neither even nor odd. x 5

The function is neither even nor odd. 82. g t

The graph displays no symmetry, which implies f x is neither odd nor even. f x

x2 4

The graph displays no symmetry, which implies f x is neither odd nor even.

z f x

79. f x

The function is even.

The function is even. 78. f x

x2 4

x 5 z f x

t 1

The graph displays no symmetry, which implies g t is neither odd nor even. g t

x 5

t 1

z g t

z f x

z g t

INSTRUCTOR USE ONLY The function is neither even nor odd. odd

The function is neither even nor odd.

NOT FOR SALE Section 1.5

83. h

top bottom 3 4 x x 3 4x x

84. h

Analyzing Graphs of Functions

4, the resulting figure is a square.

top bottom

4 x x 2 2 x 2x x2 85. L

right left 2

86. L

By the Pythagorean Theorem, 42 42 s2 Â&#x; s 32

right left

89. (a) For the average salaries of college professors, a scale of $10,000 would be appropriate.

2 0 y

(b) For the population of the United States, use a scale of 10,000,000.

2 y

87. L

4 2 meters.

0.294 x 2 97.744 x 664.875, 20 d x d 90

(a)

24 0

(b) L

2000 when x | 29.9645 | 30 watts.

88.

(b) The model is an excellent fit. (c) The temperature is increasing from 6 A.M. until noon x 0 to x 6 . Then it decreases until 2 A.M.

6 to x

until 6 A.M. x

20 . Then the temperature increases 20 to x

24 .

(d) The maximum temperature according to the model is about 63.93Â°F. According to the data, it is 64Â°F. The minimum temperature according to the model is about 33.98Â°F. According to the data, it is 34Â°F. (a) A

8 8 4

1 2

x x

Domain: 0 d x d 4

64 2 x 2

(e) Answers may vary. Temperatures will depend upon the weather patterns, which usually change from day to day.

(b)

Range: 32 d A d 64

Chapter 1

91. (a) y

(d) y

NOT FOR SALE

Functions ctions and Their Graphs

(b)

(e)

(f ) y

All the graphs pass through the origin. The graphs of the odd powers of x are symmetric with respect to the origin and the graphs of the even powers are symmetric with respect to the y-axis. As the powers increase, the graphs become flatter in the interval 1 x 1. 92. (a) Domain: > 4, 5 ; Range: >0, 9@

(b)

94. False. An odd function is symmetric with respect to the origin, so its domain must include negative values.

3, 0

(d) Relative minimum: 3, 0

95. 53 , 7

53 , 7 . (b) If f is odd, another point is 53 , 7 . (a) If f is even, another point is

Relative maximum: 0, 9

(e) Neither 93. False. The function f x

x 2 1 has a domain of

96. 2a, 2c

all real numbers.

(a)

2a, 2c

(b)

2a, 2c

97.

f x

x 2 x 4 is even.

g x

2 x 3 1 is neither.

h x

x5 2 x3 x is odd.

j x

2 x 6 x8 is even.

k x

x5 2 x 4 x 2 is neither.

p x

x 9 3 x 5 x3 x is odd.

Equations of odd functions contain only odd powers of x. Equations of even functions contain only even powers of x. A function that has variables raised to even and odd powers is neither odd nor even. 98. (a) Even. The graph is a reflection in the x-axis.

(b) Even. The graph is a reflection in the y-axis. (c) Even. The graph is a vertical translation of f. (d) Neither. The graph is a horizontal translation of f.

Section 1.6

A Library of Pare Parent Functions

Section 1.6 A Library of Parent Functions 1. f x

12. (a) f 3

axb x

2 8

f x

(h) reciprocal function 4. f x

(b)

5 4

(a) squaring function 5. f x

3 2

(b) square root function 6. f x

5 2

5 x 1

2 5 1 x 2 2

f x 2

1 x

10 4

1 3

(i) identity function 3. f x

3, 8 , 1, 2

(g) greatest integer function 2. f x

8, f 1

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019; 2 â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1

(e) constant function 7. f x

13. (a) f 5

5, 1 , 5, 1

1 1

m 5 5

(f ) absolute value function 8. f x

f x

(d) linear function

(b)

10. linear

0 10

ax b

0 x 5

y 1

1, f 5

11. (a) f 1

4, f 0

1, 4 , 0, 6

6 4 0 1

y 6

1 â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;1

2 x 0

2 x 6

f x

2 x 6

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;3

(b)

Chapter 1

Â§ 2Â· 14. (a) f Â¨ Â¸ Â© 3Â¹

NOT FOR SALE

Functions ctions and Their Graphs

15 , f 4

19. f x

x3 1

20. f x

21. f x

4 2

Â§ 2 15 Â· Â¨ , Â¸, 4, 11

2Â¹ Â©3 11 15 2

4 2 3

7 2 14 3

Â§ 7Â· Â¨ Â¸ Â© 2Â¹

f x 11

f x

Â§ 3Â· Â&#x2DC; Â¨ Â¸ Â© 14 Â¹

3 4

1 2 3

3 x 4

4 3 x 8 4

(b)

15. f x

2.5 x 4.25

16. f x

5 6

17. g x

2 x 2

18. f x

2x 3

22. h x

x 2 3

1 x

23. f x

24. k x

1 x 3

25. g x

x 5

3 x 2 1.75

NOT FOR SALE Section 1.6

26. f x

x 1

27. f x

axb

(a) f 2.1

(b) f 2.9

(d) f

A Library of Pare Parent Functions

31. g x

a xb

32. g x

4a xb

33. g x

axb 1

34. g x

ax 3b

35. g x

ÂÂ° x 6, x d 4 Â®1 Â°Ì&#x201E; 2 x 4, x ! 4

28. h x

(a)

ax 3b h 2 a1b

(b) h

12 a3.5b

1 3

(d)

a7.2b 7 h 21.6 a 18.6b

c 1 x 6f gh ed 2

29. k x

c 1 5 6f gh ed 2

(a) k 5

(b) k 6.1

a8.5b

c 1 6.1 6f gh ed 2

c 1 0.1 6f gh ed 2

(d) k 15

c 1 15 6f gh ed 2

30. g x

(a) g

a2.95b a6.05b

a13.5b

2 6

7a x 4b 6

(b) g 9

7ced 18 4fgh 6 7ced4 fhg 6 7 4 6

7 13 6

7a 4 4b 6 7a0b 6

3 2

7a9 4b 6 7a13b 6

(d) g

7 0 6

7ced 32 4fgh 6 7ced5 12 fhg 6 7 5 6

Chapter 1

36. f x

NOT FOR SALE

Functions ctions and Their Graphs

Â°Â 4 x , x 0 Â® Â°Ì&#x201E; 4 x , x t 0

2 14 x cde 14 xfhg

41. s x

(a)

(b) Domain: f, f ; Range: >0, 2

37. f x

ÂÂ°1 x 1 2 , x d 2 Â® x 2, x ! 2 Â°Ì&#x201E;

42. k x

4 12 x ced 12 xfhg

(a)

(b) Domain: f, f ; Range: >0, 4

38. f x

43. (a) W 30

14 30

420

W 40

14 40

560

W 45

21 45 40 560

665

W 50

21 50 40 560

770

x d1 Â°Âx 5, Â® 2 Â°Ì&#x201E; x 4 x 3, x ! 1 2

(b) W h

0 h d 45 Â°Â14h, Â® 21 h 45 630, h ! 45

Â°Ì&#x201E;

44. (a)

39. h x

Â4 x 2 , x 2 Â° Â®3 x, 2 d x 0 Â°x 2 1, x t 0 Â¯

The domain of f x

1.97 x 26.3 is

6 x d 12. One way to see this is to notice that this is the equation of a line with negative slope, so the function values are decreasing as x increases, which matches the data for the corresponding part of the table. The domain of f x

0.505 x 2 1.47 x 6.3 is then 1 d x d 6.

40. k x

Â2 x 1, x d 1 Â° 2 Â®2 x 1, 1 x d 1 Â°1 x 2 , x ! 1 Â¯

(b)

f 5

0.505 5 1.47 5 6.3 2

0.505 25 7.35 6.3 f 11

1.97 11 26.3

11.575

4.63

These values represent the revenue in thousands of dollars for the months of May and November, respectively. (c) These values are quite close to the actual data values.

NOT FOR SALE Section 1.7

45. Answers will vary. Sample answer:

Input Pipe

Drain Pipe 1

Drain Pipe 2

[0, 5]

Open

Closed

[5, 10]

Open

Closed

[10, 20]

Closed

[20, 30]

Closed

Open

[30, 40]

Open

[40, 45]

Open

Closed

Open

[45, 50]

Open

[50, 60]

Open

Closed

C x

Ât , 0 d t d 2 Â° ÂŽ2t 2, 2 t d 8 Â° 1 t 10, 8 t d 9 ÂŻ2

f t

To find f t

Flat fee fee per pound 26.10 4.35a xb

To find f t

y 14

1 2

1t 2

f x

(a) Domain: f, f

2t 2

10, use m

8 Â&#x; y

Total accumulation

2 and 2, 2 .

2t 2, use m

2 t 2 Â&#x; y

y 2

(b)

48. f x

47. For the first two hours the slope is 1. For the next six hours, the slope is 2. For the final hour, the slope is 12 .

Interval

46. (a) Cost

Transformations of Functions

1t 2

1 2

and 8, 14 .

14.5 inches

(a) Domain: f, f

Range: >0, f

Range: f, f

(b) x-intercept: 0, 0

y-intercept: 0, 0

Decreasing: f, 0

(d) Odd; the graph has origin symmetry.

(d) Even; the graph has y-axis symmetry. 49. False. A piecewise-defined function is a function that is defined by two or more equations over a specified domain. That domain may or may not include x- and y-intercepts.

50. False. The vertical line x 2 has an x-intercept at the point 2, 0 but does not have a y-intercept. The

horizontal line y

0, 3

3 has a y-intercept at the point

but does not have an x-intercept.

Section 1.7 Transformations of Functions 1. rigid

3. vertical stretch; vertical shrink

2. f x ; f x

4. (a) iv

(b) ii (c) iii

INSTRUCTOR USE ONLY (d) d) i

Chapter 1 5. (a) f x

NOT FOR SALE

Functions ctions and Their Graphs x c

Vertical shifts

1: f x

x 1

1 unit up

3: f x

x 3

3 units up

(b) f x

x 1

x c

1 unit down

Horizontal shifts

1: f x

x 1

1 unit right

3: f x

x 3

3 units right

6. (a) f x

x 1

x c

1 unit left

Vertical shifts

3: f x

x 3

3 units down

1: f x

x 1

1 unit down

1: f x

x 1

1 unit up

3: f x

x 3

3 units up

(b) f x

x c

Horizontal shifts

3: f x

x 3

3 units left

1: f x

x 1

1 unit left

1: f x

x 1

1 unit right

3: f x

x 3

3 units right

7. (a) f x

axb c

Vertical shifts

2: f x

2 units down

Parent function

axb 2 f x a xb f x a xb 2

ax cb

Horizontal shifts

(b) f x

2 units up

2: f x

2 units right

Parent function

ax 2b f x a xb f x a x 2b

2 units left

NOT FOR SALE Section 1.7

8. (a) f x

9. (a) y

ÂÂ°x 2 c, x 0 Â® 2 Â°Ì&#x201E; x c, x t 0

f x

(b)

(b) y

Reflection in the y-axis

(d) y

f x 4

Reflection in the x-axis and a horizontal shift 4 units to the right

(g) y

f x

Transff ormations of Functions Transformations

ÂÂ° x c 2 , x 0 Â® 2 Â°Ì&#x201E; x c , x t 0

f x 4

Vertical shift 4 units upward

(e)

f x 3

Vertical shift 3 units downward

2 f x

Vertical stretch (each y-value is multiplied by 2)

(f )

f x 1

Reflection in the x-axis and a vertical shift 1 unit downward

f 2 x

Horizontal shrink (each x-value is divided by 2)

NOT FOR SALE

Chapter 1

10. (a) y

Functions ctions and Their Graphs

f x 5

(b)

f x 3

(c)

Horizontal shift 5 units

Reflection in the x-axis and a

to the right

vertical shift 3 units upward

(d) y

f x 1

(e)

Reflection in the x-axis and a horizontal shift 1 unit to the left

(g) y

1 3

f x

Vertical shrink

each y-value is multiplied by 13

f x

(f )

Reflection in the y-axis

f x 10

Vertical shift 10 units downward

13 x

Horizontal stretch each x-value is multiplied by 3

11. Parent function: f x

13. Parent function: f x

(a) Vertical shift 1 unit downward g x

x 1 2

(b) Reflection in the x-axis, horizontal shift 1 unit to the left, and a vertical shift 1 unit upward g x

x 1 1 2

12. Parent function: f x

(a) Reflected in the x-axis and shifted upward 1 unit g x

x3 1

1 x3

x 3 1 3

g x

x 3

(b) Horizontal shift 2 units to the right and a vertical shift 4 units downward x 2 4

14. Parent function: f x

(a) Shifted downward 7 units and to the left 1 unit g x

(b) Shifted to the left 3 units and down 1 unit g x

(a) Reflection in the x-axis and a horizontal shift 3 units to the left

g x

x 1 7

(d) Reflected about the x- and y-axis and shifted to the right 3 units and downward 4 units

INSTRUCTOR USE ONLY g x

x 3 4

NOT FOR SALE Section 1.7

15. Parent function: f x

22. g x

Horizontal shift 2 units to the right y

Transff ormations of Functions Transformations

(a) Parent function: f x

16. Parent function: y

(b) Horizontal shift of 8 units to the right (c)

Vertical shrink y

1x 2

17. Parent function: f x

Reflection in the x-axis y

18. Parent function: y

(d) g x

axb

23. g x

Vertical shift y

axb 4 x

(b) Vertical shift 7 units upward

Reflection in the x-axis and a vertical shift 1 unit upward

x3 7

(a) Parent function: f x

19. Parent function: f x

f x 8

(c)

x 1

20. Parent function: y

Horizontal shift y 21. g x

x 2 12 x 2

(a) Parent function: f x

(d) g x

(b) Reflection in the x-axis and a vertical shift 12 units upward (c)

24. g x

f x 7

x3 1

(a) Parent function: f x

(b) Reflection in the x-axis, vertical shift of 1 unit downward (c)

(d) g x

12 f x

(d) g x

f x 1

NOT FOR SALE

Chapter 1

25. g x

2 x2 3

Functions ctions and Their Graphs

28. g x

(a) Parent function: f x

14 x 2 2 2

(a) Parent function: f x

(b) Vertical shrink of two-thirds, and a vertical shift 4 units upward (c)

(b) Horizontal shift 2 units to the left, vertical shrink, reflection in the x-axis, vertical shift 2 units downward (c)

(d) g x

26. g x

2 3

f x 4

2 x 7

(d) g x

(a) Parent function: f x

29. g x

(b) Vertical stretch of 2 and a horizontal shift 7 units to x2 the right of f x

(c)

(a) Parent function: f x

(b) Horizontal shrink by

x 1 3

(c)

(d) g x

27. g x

14 f x 2 2

2 f x 7

2 x 5

(d) g x

30. g x

(a) Parent function: f x

(d) g x

2 f x 5

1x 4

(a) Parent function: f x

(b) Reflection in the x-axis, horizontal shift 5 units to the left, and a vertical shift 2 units upward (c)

f 3 x

(b) Horizontal stretch of 4 (c)

(d) g x

14 x

NOT FOR SALE Section 1.7

31. g x

34. g x

1 2 3

(a) Parent function: f x

Transff ormations of Functions Transformations

12 x 1

(a) Parent function: f x

(b) Horizontal shift 1 unit to the right and a vertical shift 2 units upward

(b) Horizontal shift one unit to the right, vertical shrink

each y-value is multiplied by 12 , reflection in the

(c)

x-axis. (c)

(d) g x

32. g x

f x 1 2 (d) g x

3 10 3

(a) Parent function: f x

35. g x

12 f x 1

x 2

(b) Horizontal shift of 3 units to the left, vertical shift of 10 units downward

(a) Parent function: f x

(c)

(b) Reflection in the x-axis, vertical shift 2 units downward

(c)

(d) g x

33. g x

f x 3 10

3 x 2

(d) g x

(a) Parent function: f x

36. g x

(b) Horizontal shift 2 units to the right, vertical stretch (each y-value is multiplied by 3) (c)

f x 2

6 x 5

(a) Parent function: f x

(b) Reflection in the x-axis, horizontal shift of 5 units to the left, vertical shift of 6 units upward (c)

(d) g x

3 f x 2

(d) g x

6 f x 5

Chapter 1

37. g x

Functions ctions and Their Graphs 40. g x

x 4 8

(a) Parent function: f x

1 2

x 2 3

(a) Parent function: f x

(b) Reflection in the x-axis, horizontal shift 4 units to the left, and a vertical shift 8 units upward

(b) Horizontal shift 2 units to the right, vertical shrink, vertical shift 3 units downward

(c)

(d) g x

38. g x

f x 4 8

(d) g x

x 3 9

(a) Parent function: f x

41. g x

1 2

f x 2 3

3 a xb

(a) Parent function: f x

axb

(b) Reflection in the y-axis, horizontal shift of 3 units to the right, vertical shift of 9 units upward

(b) Reflection in the x-axis and a vertical shift 3 units upward

(c)

(d) g x

39. g x

f x 3

(d) g x

2 x 1 4

(a) Parent function: f x

42. g x

(d) g x

2 f x 1 4

2a x 5b

(a) Parent function: f x

(b) Horizontal shift one unit to the right, vertical stretch, reflection in the x-axis, vertical shift four units downward (c)

3 f x

axb

(b) Horizontal shift of 5 units to the left, vertical stretch (each y-value is multiplied by 2) (c)

(d) g x

2 f x 5

NOT FOR SALE 43. g x

x 9

(a) Parent function: f x

Section 1.7

Transff ormations of Functions Transformations

46. g x

3x 1

(a) Parent function: f x

(b) Horizontal shift 9 units to the right

(b) Horizontal shrink each x-value is multiplied by

(c)

1 3

vertical shift of 1 unit upward (c)

(d) g x

44. g x

f x 9

(d) g x

x 4 8

(a) Parent function: f x

(b) Horizontal shift of 4 units to the left, vertical shift of 8 units upward (c)

f 3 x 1

47. g x

48. g x

x 2 9

49. f x

x3 moved 13 units to the right

g x

50. f x

3 7 2

x3 moved 6 units to the left, 6 units downward,

and reflected in the y-axis (in that order) g x

(d) g x

45. g x

f x 4 8 7 x 2 or g x

(a) Parent function: f x

x 7 2

6 6 3

51. g x

x 12

52. g x

x 4 8

53. f x

(b) Reflection in the y-axis, horizontal shift 7 units to the right, and a vertical shift 2 units downward (c)

x moved 6 units to the left and reflected in

both the x- and y-axes g x

x 6

54. f x

x moved 9 units downward and reflected in

both the x-axis and the y-axis g x

55. f x

(d) g x

f 7 x 2

x 9

(a) Reflection in the x-axis and a vertical stretch (each y-value is multiplied by 3) g x

3x 2

(b) Vertical shift 3 units upward and a vertical stretch (each y-value is multiplied by 4) g x

4x2 3

Chapter 1

56. f x

Functions ctions and Their Graphs 63. Parent function: f x

(a) Vertical shrink each y -value is multiplied by g x

1 4

1 x3 4

(b) Reflection in the x-axis and a vertical stretch each y -value is multiplied by 2

g x

57. f x

2 x3

(a) Reflection in the x-axis and a vertical shrink

each y-value is multiplied by

1 2

g x

(b) Vertical stretch (each y-value is multiplied by 3) and a vertical shift 3 units downward g x

3 x 3

g x

1 2

64. Parent function: f x

g x

2 x 2

65. Parent function: f x

Reflection in the x-axis, horizontal shift 2 units to the right and a vertical shift 2 units upward g x

x 2 2 3

(a) Vertical stretch (each y-value is multiplied by 8) g x

each y-value is multiplied by 14

g x

x 4 2

67. Parent function: f x

Reflection in the x-axis and a vertical shift 3 units downward

59. Parent function: f x

g x

Vertical stretch (each y-value is multiplied by 2) 2x

Horizontal shift of 4 units to the left and a vertical shift of 2 units downward g x

(b) Reflection in the x-axis and a vertical shrink

x 3

68. Parent function: f x

Horizontal shift of 2 units to the right and a vertical shift of 4 units upward

60. Parent function: f x

Vertical stretch (each y-value is multiplied by 6) g x

each y-value is multiplied by 12

66. Parent function: f x

58. f x

g x

Reflection in the y-axis, vertical shrink

Reflection in the x-axis, vertical shift of 2 units downward, vertical stretch (each y-value is multiplied by 2)

g x

2 4 2

69. (a)

6 x

61. Parent function: f x

Reflection in the x-axis, vertical shrink

each y-value is multiplied by 12

g x

(b)

12 x 2

62. Parent function: y

Â§ x Âˇ HÂ¨ Â¸ ÂŠ 1.6 Âš

axb

Horizontal stretch (each x-value is multiplied by 2) g x

c 1 xf de 2 gh

H x

0.002 x 2 0.005 x 0.029 2

Â§ x Âˇ Â§ x Âˇ 0.002Â¨ Â¸ 0.005Â¨ Â¸ 0.029 1.6 ÂŠ Âš ÂŠ 1.6 Âš Â§ x2 Âˇ Â§ x Âˇ 0.002Â¨ Â¸ 0.005Â¨ Â¸ 0.029 ÂŠ 1.6 Âš ÂŠ 2.56 Âš 0.00078125 x 2 0.003125 x 0.029

Â§ x Âˇ The graph of H Â¨ Â¸ is a horizontal stretch of the ÂŠ 1.6 Âš graph of H x .

NOT FOR SALE Section 1.7

70. (a) The graph of N x

0.068 x 13.68 119 is 2

a reflection in the x-axis, a vertical shrink of a factor of 0.068, a horizontal shift of 13.68 units to the right and a vertical shift of 119 units upward of the graph f x

x2.

Transff ormations of Functions Transformations

76. (a) Answers will vary. Sample Answer: To graph f x 3x 2 4 x 1 use the point-plotting method

since it is not written in a form that is easily identified by a sequence of translations of the parent x2. function y (b) Answers will vary. Sample Answer: To graph f x

2 x 1 6 use the method of translating 2

the parent function y x 2 since it is written in a form such that a sequence of translations is easily identified. (b) The average rate of change from t given by the following.

3 to t

10 is

N 10 N 3

118.079 111.244 | 10 3 7 6.835 7 | 0.976

77. (a)

(b)

Each year, the number of households in the United States increases by an average of 976,000 households. (c) Let t

18:

N 18

0.068 18 13.68 119 2

(c)

| 117.7 In 2018, the number of households in the United States will be about 117.7 million households. Answers will vary. Sample answer: No, because the number of households has been increasing on average. 71. False. y

f x is a reflection in the y-axis.

72. False. y

f x is a reflection in the x-axis.

73. True. Because x

f x

x , the graphs of

x 6 and f x

x 6 are identical.

74. False. The point 2, 61 lies on the transformation. 75. y

f x 2 1

o 0 2, 1 1

o 1 2, 2 1

o 2 2, 3 1

the intervals f, 2 and 1, f

(b) Increasing on the interval 1, 2 and decreasing on the intervals f, 1 and 2, f

(d) Increasing on the interval 0, 3 and decreasing on the intervals f, 0 and 3, f

(e) Increasing on the intervals f, 1 and 4, f and

Horizontal shift 2 units to the left and a vertical shift 1 unit downward

0, 1

1, 2

2, 3

78. (a) Increasing on the interval 2, 1 and decreasing on

2, 0

1, 1

0, 2

decreasing on the interval 1, 4

79. (a) The profits were only

g t

3 4

as large as expected:

f t

(b) The profits were $10,000 greater than predicted: g t

f t 10,000

f t 2

INSTRUCTOR USE ONLY 80. No. g x

x 4 2. Yes. h x

x 3

NOT FOR SALE

Chapter 1

Functions ctions and Their Graphs

Section 1.8 Combinations of Functions: Composite Functions 1. addition; subtraction; multiplication; division 2. composition 3.

â&#x20AC;&#x201C;1

1 2

3 2

f g

6. f x

2 x 5, g x

2 x

(a)

g x

2x 5 2 x

(b)

g x

2 x 5 2 x

x 3

2x 5 2 x 3x 7 (c)

fg x

2 x

5 2 x

4 x 2 x 2 10 5 x 2 x 2 9 x 10 Â§fÂˇ (d) Â¨ Â¸ x

ÂŠgÂš

2x 5 2 x

Domain: all real numbers x except x 7. f x

(a)

x 2 , g x

g x

4x 5

f x g x

x 2 4 x 5

â&#x20AC;&#x201C;2

f g

x2 4 x 5 (b)

g x

f x g x

x 2 4 x 5

x2 4x 5

(c)

fg x

f x Â&#x2DC; g x

x 2 4 x 5

4 x3 5 x 2

Â§fÂˇ (d) Â¨ Â¸ x

ÂŠgÂš

f x

g x

x2 4x 5

5. f x

(a)

x 2, g x

g x

Domain: all real numbers x except x

x 2

5 4

f x g x

2 x 2

(b)

g x

f x g x

2 x 2

(c)

fg x

f x Â&#x2DC; g x

2 x 2

x 4 2

Â§ (d) Â¨ ÂŠ

fÂˇ Â¸ x

xÂš

f x

g x

x 2 x 2

INSTRUCTOR USE ONLY Domain: all real numbers x except xcept x

NOT FOR SALE Section on 1.8

8. f x

(a)

3 x 1, g x

g x

5x 4

Combinations of Functions: Composite Compos Composi Functions 11. f x

f x g x

3x 1 5 x 4

g x

f x g x

g x

f x g x

1 1 2 x x

x 1 x2

(b)

g x

f x g x

1 1 2 x x

x 1 x2

(c)

fg x

3 x 1 5 x 4

2 x 5

(c)

fg x

f x Â&#x2DC; g x

3 x

15 x 2 7 x 4 f x

Â§fÂ· (d) Â¨ Â¸ x

Â©gÂ¹

12. f x

4 5

Domain: all real numbers x except x 9. f x

x 2 6, g x

1 x

(a)

g x

f x g x

x2 6

(b)

g x

f x g x

x 6

(c)

fg x

f x

g x

Â§fÂ· (d) Â¨ Â¸ x

Â©gÂ¹

1 x

6 1 x

x 2 6

x2 6 1 x

x 1 2

x x3 x 1

x x 4 x3 x 1

(b)

g x

x x3 x 1

x x 4 x3 x 1

(c)

fg x

g x

x2 4

x x2 1

(b)

g x

x2 4

x2 2 x 1

Â§ x2 Â· x2 4 Â¨ 2 Â¸ Â© x 1Â¹

13.

g 2

x 1 Â&#x2DC; x 1 x3

14.

g 1

1 x 2 x 1

0 and

1 2 4

f 1 g 1

1 1 4

1 1 5

x 4 x 1 2

15.

g 0

x2 x 4 y 2 x 1 1

f 2 g 2

1 2

f 0 g 0

0 2

1 0 4

x2 4

16.

g 1

f 1 g 1

1 2

Domain: x 4 t 0 2

1 1 4

x t 4 Â&#x; x t 2 or x d 2 2

x t 2

x y x3 x 1

x4 x 1

Domain: all real numbers x except x 1 x

x Â&#x2DC; x3 x 1

Â§fÂ· (d) Â¨ Â¸ x

Â©gÂ¹

g x

Â§fÂ· (d) Â¨ Â¸ x

Â©gÂ¹

1 x

For Exercises 13â&#x20AC;&#x201C; 24, f x = x 2 + 1 and g x = x â&#x20AC;&#x201C; 4.

x 2 4, g x

fg x

x , g x

x 1

(a)

(c)

1 x

Domain: x 1 10. f x

g x

x2 x

1x 1 x2

(a)

1 x

f x Â&#x2DC; g x

f x

1 x3

Domain: all real numbers x except x

3x 1 5x 4

g x

1Â§ 1 Â· Â¨ Â¸ x Â© x2 Â¹

f x Â&#x2DC; g x

Â§fÂ· (d) Â¨ Â¸ x

Â©gÂ¹

1 5 x 4

1 x2

(a)

8x 3 (b)

1 , g x

17.

g 3t

f 3t g 3t

Âª 3t 2 1Âº 3t 4

Â¬ Â¼ 9t 2 3t 5

68 18.

Chapter 1

NOT FOR SALE

Functions ctions and Their Graphs

g t 2

f t 2 g t 2

2 1 t 2 4 2

26. f x

4 x 2 , g x

g x

4 x x 2

4 x x2

t 2 4t 4 1 t 2 4 t 2 3t 1

19.

fg 6

f 6 g 6

1 6 4

20.

fg 6

f 6 Â&#x2DC; g 6

ÂŞ 6 2 1Âş ÂŞ 6 4Âş Âź ÂŹ ÂźÂŹ

37 10

370

f 5

52 1 5 4

f 0

02 1 0 4

Â§fÂˇ 23. Â¨ Â¸ 1 g 3

ÂŠgÂš

f 1

Â§fÂˇ 21. Â¨ Â¸ 5

ÂŠgÂš

g 5

Â§fÂˇ 22. Â¨ Â¸ 0

ÂŠgÂš

g 0

2 1 5 f 4

x3 10

magnitude. For x ! 6, g x contributes most to the magnitude.

3 4

3 5

f 5 g 5 f 4

g x

1 4

g 3

1 4

fg 5

3 x, g x

For 0 d x d 2, f x contributes most to the

g 1

1 2

24.

27. f x

x , g x

2 x f g x

28. f x

x x

1 5 4 42 1

26 Â&#x2DC; 1 17 43

25. f x

1 x, 2

g x

3 x 2

x 1 1

g x contributes most to the magnitude of the sum for 0 d x d 2. f x contributes most to the magnitude of the sum for x ! 6. 29. f x

3x 2, g x

g x

x 5

x 5 2

For 0 d x d 2, f x contributes most to the magnitude. For x ! 6, f x contributes most to the magnitude.

NOT FOR SALE Section on 1.8

30. f x

x 2 12 , g x

g x

Combinations of Functions: Composite Compos Composi Functions

3x 2 1

2 x 2

34. f x

For 0 d x d 2, g x contributes most to the magnitude. For x ! 6, g x contributes most to the magnitude. 31. f x

x 2 , g x

(a)

D g x

f g x

Â§1Âˇ fÂ¨ Â¸ ÂŠ xÂš

Â§1Âˇ Â¨ Â¸ ÂŠ xÂš

(b)

D f x

g f x

g x3

1 x3

(c)

D g x

g g x

Â§1Âˇ gÂ¨ Â¸ ÂŠ xÂš

35. f x

g x

x 1

(a)

D g x

f g x

f x 1

(b)

D f x

g f x

g x2

(c)

D g x

g g x

g x 1

(a)

3 x 5, g x

D g x

(b)

5 x

Domain: all real numbers x

D g x

(b)

D f x

3 5 x 5

36. f x

20 3x

g x

g f x

(a)

g 3 x 5

D f x

33. f x

(a)

D g x

x 1, g x

D g x

D f x

g 5 x

(b)

x3 1 1

x (c)

D g x

37. f x

g x

1 1

D f x

(a)

x2 1 x

D g x

x3 1 5

x3 4

g f x

x 5

1 x 4

Domain: all real numbers x Domain: x t 0 f g x

x 1 f

g g x

g x3 1

x3 1

Domain: all real numbers x

x 1

x 4

f g x

x 5 1

x 1

Domain: all real numbers x

g f x

Domain: all real numbers x

x 4

Domain: all real numbers x

x3 1

f g x

x 4

f x3 1

(b)

x3 1

5 3x 5

g g x

x2 4

g f x

x 5

3x (c)

f x2

f g x

Domain: x t 4

f g x

f 5 x

1 x3

Domain: x t 4

x 4

32. f x

Domain: all real numbers x

x2 1 x 2

1 x

x 3 , g x

3 2

x 1

x9 3x6 3x3 2

Domain: x t 0 (b)

D f x

g f x

g x 2 1

x2 1

INSTRUCTOR USE ONLY Domain: all real numbers x

Chapter 1

38. f x

g x

(a)

NOT FOR SALE

Functions ctions and Their Graphs

x2 3

Domain: all real numbers x

39. f x

Domain: all real numbers x

g x

D g x

f x6

f g x

(a)

D f x

g x2 3

g f x

x2 3

(b)

40. f x

x 4

Domain: all real numbers x

g x

3 x

Domain: all real numbers x

D g x

f g x

x 6

Domain: all real numbers x

D g x

f g x

f x 6

x 6

D f x

g f x

g x

x 6

Domain: all real numbers x

(a)

Domain: all real numbers x

Domain: all real numbers x (b)

f 3 x

3 x

x 1

Domain: all real numbers x (b)

D f x

g x 4

g f x

x 4

3 x 4

Domain: all real numbers x

41. f x

g x

(a)

1 x

Domain: all real numbers x except x

x 3

Domain: all real numbers x

D g x

f g x

1 x 3

f x 3

Domain: all real numbers x except x (b)

D f x

Â§1Âˇ gÂ¨ Â¸ ÂŠ xÂš

g f x

g x

(a)

43. (a)

Domain: all real numbers x except x

x 1

Domain: all real numbers x

D g x

D f x

f g x

f x 1

(b)

Â§ 3 Âˇ gÂ¨ 2 Â¸ ÂŠ x 1Âš

g f x

1 1

0 and x

g 3

g 1

fg 4

f 3 g 3

f 2

g 2

0 2

f 1 g 1

f 4 Â&#x2DC; g 4

2 1

2 3 4Â&#x2DC;0

3 x2 2x

3 1 x 1 2

3 x2 2 x 1 1

Domain: all real numbers x except x

Â§fÂˇ (b) Â¨ Â¸ 2

ÂŠgÂš

44. (a)

3 x 1 2

Domain: all real numbers x except x (b)

1 3 x

Domain: all real numbers x except x

42. f x

3 x2 1 x2 1

x2 2 x2 1

45. (a)

D g 2

f g 2

f 2

(b)

D f 2

g f 2

g 0

46. (a)

D g 1

f g 1

f 3

(b)

D f 3

g f 3

g 2

NOT FOR SALE Section on 1.8

47. h x

2 x2

48. h x

x and g x

2 x 1,

h x .

One possibility: Let g x

then f D g x

49. h x

x and g x

One possibility: Let g x

h x .

54. h x

4 , x2

x 3 and g x

4 x

x2 ,

h x .

27 x3 6 x 10 27 x3

One possibility: Let g x

9 x and f x

5 x 2 and f x

x2 3 4 x2

53. h x

then f D g x

h x .

f x

55. (a) T x

1 x 2

x3 and

27 x 6 3 x , then f D g x

10 27 x R x B x

3 x 4

h x .

1 x2 15

(b)

One possibility: Let f x

then f D g x

x 2 4,

9 x

then f D g x

One possibility: Let f x

x 4

One possibility: Let g x

51. h x

x3 ,

h x .

One possibility: Let f x

50. h x

1 x and f x

then f D g x

5 x

then f D g x

1 x 3

52. h x

One possibility: Let f x

then f D g x

Combinations of Functions: Composite Compos Composi Functions

1 x and g x

x 2,

h x .

56. (a) c t

b t d t

p t

u 100

(b) c(5) represents the percent change in the population due to births and deaths in the year 2005.

57. (a) p t

d t c t

(b) p(5) represents the number of dogs and cats in 2005. (c) h t

p t

n t

d t c t

n t

h t represents the number of dogs and cats at time t compared to the population at time t or the number of dogs and cats per capita.

58. (a) T is a function of t since for each time t there corresponds one and only one temperature T. (b) T 4 | 60q; T 15 | 72q (c) H t

T t 1 ; All the temperature changes would be one hour later.

(d) H t

T t 1; The temperature would be decreased by one degree.

NOT FOR SALE

Chapter 1

Functions ctions and Their Graphs

(e) The points at the endpoints of the individual functions that form each â&#x20AC;&#x153;pieceâ&#x20AC;? appear to be 0, 60 , 6, 60 , 7, 72 , 20, 72 , 21, 60 , and 24, 60 . Note that the value t 24 is chosen for the last ordered pair because that is when the day ends and the cycle starts over. From t

0 to t

6: This is the constant function T t

From t

6 to t

7: Use the points 6, 60 and 7, 72 .

72 60 12 7 6 y 60 12 x 6 Â&#x; y

60.

From t

7 to t

From t

20 to t

12 x 12, or

T t

12t 12

20: This is the constant function T t

72.

21: Use the points 20, 72 and 21, 60 .

72 60 12 20 21 12 x 21 Â&#x; y y 60

From t

21 to t

12 x 312, or T t

24: This is the constant function T t

A piecewise-defined function is T t

59. (a) r x

(b) A r

(c)

Â60, Â° Â°12t 12, Â° ÂŽ72, Â° Â° 12t 312, Â°ÂŻ60,

60.

0 d t d 6 6 t 7 7 d t d 20 20 t 21 21 d t d 24

61. (a) f g x

x 2

Â§ xÂˇ AÂ¨ Â¸ ÂŠ 2Âš

A r x

A D r x

Â§ xÂˇ ÂŠ Âš

represents the area of the circular base of

62. (a) R p

10 9t 2 12t 4 60t 40 600

(c)

90t 2 60t 600

3t 2 2t 20

0.9 p 2000

S p 2000

(d)

3t 2 2t 30 0 By the Quadratic Formula, t | 3.513 or 2.846.

D S p represents the factory rebate after the D R p represents the dealership discount after D S p

D S 20,500

0.9 20,500 2000

0.9 p 1800

the factory rebate.

652.5

1500

Choosing the positive value for t, you have t | 2.846 hours.

R 0.9 p

dealership discount.

After half an hour, there will be about 653 bacteria. (c) 30 3t 2 2t 20

R D S p

S D R p

This represents the number of bacteria in the food as a function of time. 30 3 0.5 2 0.5 20

0.9 p the cost of the car with the dealership

0.9 p 2000

30 3t 2 2t 20 , 0 d t d 6

N T 0.5

p 2000 the cost of the car after the

discount.

0.03 x 500,000

factory rebate. (b) S p

N 3t 2

0.5.

g x 500,000

over $500,000.

10 3t 2 20 3t 2 600

(b) Use t

0.03 x 500,000

g f x

represents your bonus of 3% of an amount

SÂ¨ Â¸ 2

the tank on the square foundation with side length x.

60. (a) N T t

f 0.03 x

(b) g f x

Sr2

A D r x

12t 312

D R p

D R 20,500

0.9 20,500 1800

$16,450 $16,650

D S 20,500 yields the lower cost because

10% of the price of the car is more than $2000

NOT FOR SALE Section on 1.8

63. Let O

oldest sibling, M

middle sibling, Y

Combinations of Functions: Composite Compos Composi Functions

youngest sibling.

Then the ages of each sibling can be found using the equations: O

1Y 2

(a) O M Y

12 Y 6

12 Y ; Answers will vary.

(b) Oldest sibling is 16: O

Middle sibling: O

8 years old

Youngest sibling: M

1Y 2

1 Y 2

1Y 2

4 years old

64. (a) Y M O

2 12 O 12

O 12; Answers will vary.

(b) Youngest sibling is 2 o Y 1Y 2

Middle sibling: M

1 2

7 years old

Oldest sibling: O

65. False. f D g x

2 7

14 years old

6 x 1 and g D f x

6x 6

66. True. The range of g must be a subset of the domain of f for f D g x to be defined.

68. (a) f p : matches L 2 ; For example, an original price of p S

(b) g p : matches L1; For example an original price of

67. Let f x and g x be two odd functions and define h x

f x g x . Then

h x

f x g x

(d)

h x .

Let f x and g x be two even functions and define h x

f x g x . Then

h x

f x g x

So, h x is even.

D f p : matches L 4 ; This function represents

D g p matches L 3 ; This function represents

subtracting a $5 discount from the original price p, then applying a 50% discount.

So, h x is even.

h x .

$20.00 corresponds to a sale price of $15.00.

applying a 50% discount to the original price p, then subtracting a $5 discount.

because f and g are odd

f x g x

p S (c)

ÂŞ ÂŹ f x ÂşÂŞ ÂźÂŹ g x ÂşÂź

$15.00 corresponds to a sale price of $7.50.

69. Let f x be an odd function, g x be an even function, and define h x

h x

because f and g are even

f x g x . Then

f x g x

ÂŞ ÂŹ f x ÂşÂź g x

because f is odd and g is even

f x g x

h x .

So, h is odd and the product of an odd function and an even ven function is odd.

NOT FOR SALE

Chapter 1

70. (a) g x

Functions ctions and Their Graphs 1 Âª f x f x ÂºÂ¼ 2Â¬

(c)

To determine if g x is even, show g x

g x .

f x

x2 2 x 1

f x

g x h x

g x

1 Âª f x f x ÂºÂ¼ 2Â¬ 1Âª 2 2 x 2 x 1 x 2 x 1Âº Â¼ 2Â¬ 1 2 2 Âª x 2 x 1 x 2 x 1ÂºÂ¼ 2Â¬ 1 Âª2 x 2 2ÂºÂ¼ x2 1 2Â¬ 1 Âª f x f x ÂºÂ¼ 2Â¬ 1Âª 2 2 x 2 x 1 x 2 x 1 ÂºÂ» Â¼ 2 Â¬Â« 1 2 2 Âª x 2 x 1 x 2 x 1ÂºÂ¼ 2Â¬ 1 > 4 x@ 2 x 2

1 Âª f x f x

ÂºÂ¼ 2Â¬ 1 Âª f x f x ÂºÂ¼ 2Â¬ 1 Âª f x f x ÂºÂ¼ 2Â¬

g x

g x 9

h x

1 Âª f x f x ÂºÂ¼ 2Â¬

h x

To determine if h x is odd show h x

h x

(b) Let f x

f x

h x .

1 Âª f x f x

ÂºÂ¼ 2Â¬ 1 Âª f x f x ÂºÂ¼ 2Â¬ 1 ÂªÂ¬ f x f x ÂºÂ¼ 2

f x

k x

h x 9

k x

1 x 1 g x h x

g x

a function even function odd function.

Using the result from part (a) g x is an even function and h x is an odd function. f x

g x h x

1 2 x

1 Âªk x k x ÂºÂ¼ 2Â¬ 1Âª 1 1 Âº 2 Â«Â¬ x 1 x 1Â»Â¼ 1 Âª1 x x 1Âº 2 Â«Â¬ x 1 1 x Â»Â¼ 1Âª 2 Âº 2 Â«Â¬ x 1 1 x Â»Â¼ 1 x 1 1 x

1 1 Âª f x f x ÂºÂ¼ ÂªÂ¬ f x f x ÂºÂ¼ 2Â¬ 2 1 1 1 1 f x f x f x f x

2 2 2 2 f x 9

x h x

1 1 x 1

1 Âªk x k x ÂºÂ¼ 2Â¬ 1Âª 1 1 Âº Â« 2 Â¬ x 1 1 x Â»Â¼ 1 Âª1 x x 1 Âº Â« Â» 2 Â«Â¬ x 1 1 x Â»Â¼ Âº 2x 1Âª Â« Â» 2 Â¬Â« x 1 1 x Â»Â¼ x x 1 1 x

x k x

x 1 x 1

NOT FOR SALE Section 1.9

Inver Inverse Functions

Section 1.9 Inverse Functions 9. f x

1. inverse

3x 1

x 1 3 Â§ x 1Â· f f 1 x

fÂ¨ Â¸ Â© 3 Â¹

f 1 x

2. f 1 3. range; domain 4. y

f 1 f x

Â§ x 1Â· 3Â¨ Â¸ 1 x Â© 3 Â¹ 3x 1 1

f 1 3 x 1

5. one-to-one 10. f x

6. Horizontal 7. f x

x 6

f 1 x

f f 1 x

f 1 f x

8. f x

f 1 x

6x 1 x 6 Â§ xÂ· fÂ¨ Â¸ Â©6Â¹

Â§ xÂ· 6Â¨ Â¸ Â©6Â¹ 6x f 1 6 x

f x

3 x

1 1

f 5 x 1

f 1 f x

Â§ x 1Â· Â§ x 1Â· 5Â¨ f 1 Â¨ Â¸ Â¸ 1 Â© 5 Â¹ Â© 5 Â¹ x 1 1 x

1x 3

f x3

f 1 f x

f 1

12. f x

14.

15.

16.

D g x

f g x

Â§ 2x 6 Â· f Â¨ Â¸ 7 Â¹ Â©

D f x

g f x

Â§ 7 Â· g Â¨ x 3Â¸ 2 Â© Â¹

D g x

f g x

f 4 x 9

D f x

g f x

Â§ x 9Â· gÂ¨ Â¸ Â© 4 Â¹

D g x

f g x

D f x

g f x

g x3 5

D g x

f g x

D f x

g f x

Â§ x3 Â· gÂ¨ Â¸ Â©2Â¹

x 5

x3 5 5 2x

x 3

x5 5

x x

f 1 f x

f 1 x5

x 3 3 7 x

x x

2x 2

2 Â§ x3 Â· 2Â¨ Â¸ Â©2Â¹

x 5 5

x x

f f 1 x

4x 9 9 4x x 4 4 Â§ x 9Â· x 9 9 4Â¨ Â¸ 9 Â© 4 Â¹

x 5

7 Â§ 2x 6 Â· Â¨ Â¸ 3 2Â© 7 Â¹ Â§ 7 Â· 2Â¨ x 3Â¸ 6 2 Â¹ Â© 7

f 1 x

13.

5x 5

x x3 f f 1 x

3 x

5x 1 1 5

f f 1 x

11. f x

1 3

x 1 5 5x 1

1x 3

x 3x f f 1 x

f 3 x

INSTRUCTOR C USE ONLY 3

NOT FOR SALE

Chapter 1

17.

Functions ctions and Their Graphs

22. f x

y 3

f x 5

g f x

g x 5

1 â&#x2C6;&#x2019;1

x 1

x 5

(a) f g x

â&#x2C6;&#x2019;3

x 5, g x

x 5 5 x 5 5

x x

(b)

â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;3

18. 7 6 5 4 3

23. f x

7 x 1, g x

1 x

â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1

(a) f g x

Â§ x 1Â· fÂ¨ Â¸ Â© 7 Â¹

g f x

g 7 x 1

19. 4

x 1 7 Â§ x 1Â· 7Â¨ Â¸ 1 Â© 7 Â¹

7 x

1 1 7

x x

(b)

3 2 1 x

â&#x2C6;&#x2019;1

20. 3

24. f x

3 4 x, g x

1 â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;2

(a) f g x

â&#x2C6;&#x2019;1

Â§3 xÂ· fÂ¨ Â¸ Â© 4 Â¹ 3 3 x

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;3

21. f x

3 x 4

g f x

2 x, g x

x 2

(a) f g x

Â§ xÂ· fÂ¨ Â¸ Â© 2Â¹

g f x

g 2 x

g 3 4 x

Â§3 xÂ· 3 4Â¨ Â¸ Â© 4 Â¹ x 3 3 4 x

4x 4

(b) Â§ xÂ· 2Â¨ Â¸ x Â© 2Â¹ 2x x 2

(b)

NOT FOR SALE Section 1.9

25. f x

x3 , g x

(a) f g x

g f x

Â§ x3 Â· gÂ¨ Â¸ Â©8Â¹

27. f x

8 3

x 4, g x

(a) f g x

Â§ x3 Â· 8Â¨ Â¸ Â©8Â¹

8x 8

g f x

x 2 4, x t 0

f x 2 4 , x t 0

x2 g

4 4

x 4

(b)

Inver Inverse Inve Functions

x 4

(b)

26. f x

1 , g x

1 x

28. f x

(a) f g x

Â§1Â· fÂ¨ Â¸ Â© xÂ¹

1 1x

1 x

1Â&#x2DC;

x 1

g f x

Â§1Â· gÂ¨ Â¸ Â© xÂ¹

1 1x

1 x

1Â&#x2DC;

x 1

1 x 3 , g x

(a) f g x

1 x

1 1 x

g f x

(b)

g 1 x3

1 x

x 3

1 1 x3

(b)

29. f x

9 x 2 , x t 0; g x

(a) f g x

g f x

9 x, x d 9

9 x ,x d 9

g 9 x 2 , x t 0

9 x

9 9 x 2

x x

(b)

Chapter 1

30. f x

NOT FOR SALE

Functions ctions and Their Graphs

1 , x t 0; g x

1 x

(a) f g x

Â§1 x Â· fÂ¨ Â¸ Â© x Â¹

g f x

Â§ 1 Â· gÂ¨ Â¸ Â©1 x Â¹

1 x ,0 x d 1 x

1 Â§1 x Â· 1 Â¨ Â¸ Â© x Â¹

1 x 1 x x x

1 1 x

Â§ 1 Â· 1 Â¨ Â¸ Â©1 x Â¹ Â§ 1 Â· Â¨ Â¸ Â©1 x Â¹

1 x 1 1 x 1 x 1 1 x

x 1 x 1 1 x

x x 1 Â&#x2DC; 1 x 1

(b)

31. f x

x 1 , g x

x 5

(a) f g x

g f x

5x 1 x 1

Â§ 5x 1Â· f Â¨ Â¸ Â© x 1Â¹

Â§ x 1Â· gÂ¨ Â¸ Â© x 5Â¹

Â§ 5x 1 Â· 1Â¸ Â¨ x 1 Â© Â¹ Â&#x2DC; x 1 Â§ 5x 1 Â· x 1 5Â¸ Â¨ 1 x Â© Â¹ Âª Â§ x 1Â· Âº Â«5Â¨ x 5 Â¸ 1Â» x 5 Â© Â¹ Â¼ Â&#x2DC; Â¬ Âªx 1 Âº x 5 Â«Â¬ x 5 1Â»Â¼

5 x 1 x 1

6x 6

5 x 1 5 x 1

5 x 1 x 5

x 1 x 5

6x 6

(b)

NOT FOR SALE Section 1.9

x 3 , g x

x 2

32. f x

2x 3 3 x 1 2x 3 2 x 1 Â§ x 3Âˇ 2Â¨ Â¸ 3 ÂŠ x 2Âš x 3 1 x 2

Â§ x 3Âˇ gÂ¨ Â¸ ÂŠ x 2Âš

g f x

2x 3 x 1

Â§ 2x 3 Âˇ fÂ¨ Â¸ ÂŠ x 1Âš

(a) f g x

Inver Inverse Functions

2 x 3 3x 3 x 1 2x 3 2x 2 x 1 2x 6 x x 3 x

3x 6 2 x 2 2

5x 5

(b)

^ 2, 1 , 1, 0 , 2, 1 , 1, 2 , 2, 3 , 6, 4 `

33. No,

does

42. f x

not represent a function. 2 and 1 are paired with two different values.

34. Yes,

1 8

2 1 2

^ 10, 3 , 6, 2 , 4, 1 , 1, 0 , 3, 2 , 10, 2 `

does represent a function.

35.

x f 1 x

36.

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;1

f does not pass the Horizontal Line Test, so f does not have an inverse.

43. f x

x f

â&#x20AC;&#x201C;10

â&#x20AC;&#x201C;7

â&#x20AC;&#x201C;4

â&#x20AC;&#x201C;1

â&#x20AC;&#x201C;3

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;1

2 x 16 x 2

37. Yes, because no horizontal line crosses the graph of f at more than one point, f has an inverse. 38. No, because some horizontal lines intersect the graph of f twice, f does not have an inverse. 39. No, because some horizontal lines cross the graph of f twice, f does not have an inverse.

f does not pass the Horizontal Line Test, so f does not have an inverse.

44. h x

x 4 x 4

40. Yes, because no horizontal lines intersect the graph, of f at more than one point, f has an inverse. 41. g x

h does not pass the Horizontal Line Test, so h does not have an inverse.

INSTRUCTOR USE ONLY g passes the Horizontal Line Test, so g has an inverse.

Chapter 1

45. (a)

NOT FOR SALE

Functions ctions and Their Graphs

f x

2x 3

2y 3

(b)

x 3 2 x 3 2

y f 1 x

(d) The domains and ranges of f and f 1 are all real numbers. f x

3x 1

3y 1

x 1 3

f 1 x

x 1 3

x5 2

x 2

y5 2

x 2

4 y2

4 x2 4 x2

f 1 x

4 x2 , 0 d x d 2

(b)

(d) The domains and ranges of f and f 1 are all real numbers x such that 0 d x d 2. f x

50. (a) are all real

(b)

f 1 x

4 x2

(d) The domains and ranges of f and f numbers. f x

(b)

47. (a)

4 x2 , 0 d x d 2

46. (a)

f x

49. (a)

x 2 2, x d 0

x2 2

y2 2

x 2

f 1 x

x 2

(b)

(c) The graph of f 1 is the reflection of the graph of f x. in the line y (d) The domains and ranges of f and f 1 are all real numbers.

48. (a)

f x

x 1

y3 1

x 1

(d) > 2, f is the range of f and domain of f 1.

x 1 1

(b)

f, 0@

is the domain of f and the range of f 1.

x 1

(c) The graph of f 1 is the reflection of f in the line y x. (d) The domains and ranges of f and f 1 are all real numbers.

NOT FOR SALE Section 1.9

4 x 4 x 4 y

f x

51. (a)

y x xy y f

f x

54. (a)

(b)

y x

xy 2 x y 3

4 x 4 x

y x 1

y f 1 x

(c) The graph of f 1 is the same as the graph of f. (d) The domains and ranges of f and f 1 are all real numbers except for 0.

52. (a)

2 x 2 x 2 y

f x

y x

x 1 x 2 x 1 x 2 y 1 y 2

x y 2

y 1

xy 2 x

y 1

xy y

2x 1

y x 1

2x 1

y f 1 x

0 2x 3 2x 3 x 1 2x 3 x 1

(d) The domain of f and the range of f 1 is all real 2. numbers except x The range of f and the domain of f 1 is all real numbers x except x 1.

(d) The domains and ranges of f and f 1 are all real numbers except for 0. f x

3 2 3 2 3 2

(b)

53. (a)

2 x 2 x

f 1 x

x x x x y y

(b)

Inver Inverse Functions

55. (a) (b)

f x

x 1

y 1

x3 1

f 1 x

x3 1

(b)

(c) The graph of f 1 is the reflection of the graph of f in the line y x. (d) The domains and ranges of f and f 1 are all real numbers.

2x 1 x 1 2x 1 x 1

(c) The graph of f 1 is the reflection of graph of f in the line y x. (d) The domain of f and the range of f 1 is all real numbers except 2. The range of f and the domain of f 1 is all real numbers except 1.

NOT FOR SALE

Chapter 1

56. (a)

Functions ctions and Their Graphs

f x

x3 5

y3 5

x5 3

y3 5

x5 3

f 1 x

60.

(b)

f x

3x 5

3y 5

x 5

x 5 3

This is a function of x, so f has an inverse.

x5 3

(c) The graph of f 1 is the reflection of the graph of f in the line y x. (d) The domains and ranges of f and f 1 are all real numbers.

61. p x

r4 x

and fails the Horizontal Line Test. p does not have an inverse.

62.

x y

y x

1 x2 1 x2 1 y2 1 x

5x 5x 4 5x 4 3

y x y

4 4

3y y

63. f x

3 , x t 3 Â&#x; y t 0

3 , x t 3, y t 0

3 , y t 3, x t 0

y 3, y t 3, x t 0

x 8 x 8 y 8 8x

2 2

x 3, x t 0, y t 3

This is a function of x, so f has an inverse. f 1 x

This is a function of x, so g has an inverse. g 1 x

5x 4 3

f 1 x

This does not represent y as a function of x. f does not have an inverse.

59. g x

This is a function of x, so f has an inverse.

1 x

3x 5 3x 5 3y 5 3y

f x

This does not represent y as a function of x. f does not have an inverse.

4 for all x, the graph is a horizontal line

Because y

57. f x

58. f x

x 5 3

f 1 x

x 3, x t 0

q x

y 5

64.

This does not represent y as a function of x, so q does not have an inverse.

NOT FOR SALE Section 1.9

Âx 3, x 0 ÂŽ ÂŻ6 x, x t 0

65. f x

Inver Inverse Functions

3 2x 3 Â&#x; x t , y t 0 2 3 2 x 3, x t , y t 0 2 3 2 y 3, y t , x t 0 2 3 2 y 3, x t 0, y t 2 2 x 3 3 , x t 0, y t 2 2

69. f x

y x x2 y

This is a function of x, so f has an inverse. This graph fails the Horizontal Line Test, so f does not have an inverse. x d 0 Â x, ÂŽ 2 ÂŻ x 3 x, x ! 0

66. f x

x2 3 ,x t 0 2

f 1 x

f x

70.

x 2 Â&#x; x t 2, y t 0

x 2, x t 2, y t 0

y 2, y t 2, x t 0

y 2, x t 0, y t 2

x 2

y, x t 0, y t 2

This is a function of x, so f has an inverse. f 1 x

The graph fails the Horizontal Line Test, so f does not have an inverse.

71.

x 2 2, x t 0

f x

4 2 x

67. h x

x y

2 x

â&#x2C6;&#x2019;4

4 â&#x2C6;&#x2019;2

68. f x

x 2,x d 2 Â&#x; y t 0

x 2 , x d 2, y t 0

y 2 , y d 2, x t 0

x 2 x

y 2 or y

or 2 x

4 5 4 5 4 5

x 4 y 5

6y 4

4 xy 5 x

6y 4

4 xy 6 y

5x 4

y 4 x 6

5x 4

The graph fails the Horizontal Line Test so h does not have an inverse.

6x 4x 6x 4x 6y 4y

5x 4 4x 6 5x 4 6 4x

This is a function of x, so f has an inverse. f 1 x

5x 4 6 4x

y 2 y

The portion that satisfies the conditions y d 2 and y. This is a function of x, so f has x t 0 is 2 x an inverse.

INSTRUCTOR USE ONLY f 1 x

2 x, x t 0

NOT FOR SALE

Chapter 1

Functions ctions and Their Graphs

72. The graph of f passes the Horizontal Line Test. So, you know f is one-to-one and has an inverse function. 5x 2x 5x 2x 5y 2y

f x

y x

3 5 3 5 3 5

75. f x

x 2

domain of f : x t 2, range of f : y t 0 f x

x 2

y 2

x 2

x 2 y 5

5y 3

So, f 1 x

2 xy 5 x

5y 3

domain of f 1 : x t 0, range of f 1 : y t 2

2 xy 5 y

5x 3

y 2 x 5

5 x 3

5x 2x 5x 2x

y f 1 x

73. f x

76. f x

y 2

x 5

y 5

So, f

domain of f : x t 0, range of f : y d 1

1 x4

1 y4

x 1

1 x

So, f 1 x

f x

y 6

x 6

f x

x 5.

domain of f : x t 6, range of f : y t 0

x 2.

1 x

77. f x

domain of f 1 : x t 0, range of f 1 : x t 2

74. f x

domain f 1 : x t 0, range of f 1 : y t 5

So, f 1 x

f x

x 5

domain of f : x t 2, range of f : y t 0

x 2

x 5

domain of f : x t 5, range of f : y t 0

3 5 3 5

f x

x 2.

So, f

x 6.

domain of f 1 : x t 0, range of f 1 : y t 6

78. f x

domain of f : x t 4, range of f : y t 0 f x

y 4

1 x.

domain of f 1 : x d 1, range of f 1 : y t 0

x 4

So, f

x 4.

domain of f 1 : x t 0, range of f 1 : y t 4

NOT FOR SALE Section 1.9

79. f x

82. f x

2 x2 5

2x2 5

f x

x 1 2

2x 5

x 1 2

2y 5

y 1 2

x 5

y 1

5 x

2 y2

5 x 2

1 x2 2

g x = x 3 , g â&#x20AC;&#x201C;1 x =

2 x 5

x : x

83.

d 5, range of f

x : y

domain of f : x t 0, range of f : y t 1

f x

1 2 x 2

1 x2 2

1 y2 2

x 1

1 2 y 2

2x 2

So, f

domain of f

81. f x

f 1 g 1 1

f 1

1 3

84.

g 1 D

f 1 3

85.

f 1 D

f 1 6

8ÂªÂ¬8 6 3 3ÂºÂ¼

86.

g 1 D g 1 4

:y t 0

1 x3 8

3 3

y 3

x 3

1 3 y 8

y 3

:y t 4

8 x 3

D g

23 x 3

y x

f g x

D g x

x 3

x 3.

3 3

87.

600

g 1 g 1 4

g 1

: x t 1, range of f

f 1 f 1 6

domain of f

f 1 8>6 3@

x 4 1

g 1 8 3 3

x 4 1

So, f 1 x

g 1 f 1 3

1 3 y 8

x 3

g 1 0

domain of f : x t 4, range of f : y t 1 f x

f 1 D g 1 1

â&#x20AC;&#x201C; 3, f â&#x20AC;&#x201C;1 x = 8 x + 3 ,

1x 8

2 x 2. 1

t 0

2x 2

x 1.

In Exercises 83â&#x20AC;&#x201C; 88, f x =

So, f 1 x

domain of f

So, f 1 x

x 1

domain of f 1 : x d 2, range of f 1 : y t 1

2 2

2 5 x

x 1 2

f x

5 x Â&#x2DC; 2

domain of f : x t 1, range of f : y d 2

domain of f : x t 0, range of f : y d 5

80. f x

Inver Inverse Inve Functions

f x3

1 x3 8

NOT FOR SALE

Chapter 1

Functions ctions and Their Graphs g 1 f 1 x

88. g 1 D f 1

93. (a)

g 1 8 x 3

8 x 3

23 x 3

89.

D f

x 10 0.75

f x

(b) y

90.

D g

4 5

91.

24.25 10 0.75

number of units produced 19

0.03x 2 245.50, 0 x 100 Â&#x; 245.50 y 545.50

g x

5Âˇ Â¸ 2 Âš 4

0.03 y 2 245.50

x 245.50

0.03 y 2

x 245.50 0.03

y, 245.50 x 545.50

f 1 x

x 245.50 0.03

temperature in degrees Fahrenheit

percent load for a diesel engine

(b)

f g x

D g x

hourly wage, y

94. (a)

Â§x f 1 Â¨ ÂŠ x 5 2 x 5 2 x 3 2

x 10 . 0.75

So, 19 units are produced.

2 x 1 2 1

10 0.75 y 0.75 y

g 1 x 4

x x 10

x +5 . 2 1

10 0.75 x

So, f 1 x

In Exercises 89â&#x20AC;&#x201C;92, f x = x + 4, f â&#x20AC;&#x201C;1 x = x â&#x20AC;&#x201C; 4, g x = 2 x â&#x20AC;&#x201C; 5, g â&#x20AC;&#x201C;1 x =

f 2 x 5

2 x

5 4

2x 1

D g

x 1 2

0.03 x 2 d 254.50 x 2 d 8483.33

Note: Comparing Exercises 89 and 91,

f 92.

D g

D f x

g 1 D

f 1 x .

g f x

g x 4

2 x 4 5 2x 8 5 2x 3

2x 3

2y 3

x 3

x 3 2

x d 92.10

Thus, 0 x d 92.10. 95. False. f x

x 2 is even and does not have an inverse.

96. True. If f x has an inverse and it has a y-intercept at

0, b ,

then the point b, 0 , must be a point on the

graph of f 1 x .

x 3 2

INSTRUCTOR USE ONLY g

D f

NOT FOR SALE Section 1.9

97.

101. If f x

f 1 3

x f 1 x

2, then f 2

k 2 2 8

12k

3 3 12

1. 4

f f 1 x

â&#x20AC;&#x201C;10

f x

â&#x20AC;&#x201C;10

1. 4

98.

3. So,

k 2 2 2

102.

k 2 x x3 has an inverse and

f 2

So, k

Inver Inverse Inve Functions

f x and f 1 x are inverses of each other. 103.

The graph does not pass the Horizontal Line Test, so f 1 x does not exist. 99. Let f D g x

D g x

y. Then x y Â&#x; f g x

g x

x x

D g

y .

Also,

y f

Because f and g are both one-to-one functions, f D g

g 1 D f 1.

f x . Letting x, y be any point

on the graph of f x Â&#x; x, y is also on the graph of f x and f 1 y

also an odd function.

and the

range of f is equal to the domain of f . 104. (a) C x is represented by graph m and C 1 x is represented by graph n.

(b) C x represents the cost of making x units of personalized T-shirts. C 1 x represents the number

100. Let f x be a one-to-one odd function. Then f 1 x

exists and f x

x 1 because

the domain of f is equal to the range of f 1

f y

g 1 D f 1 y . g

There is an inverse function f 1 x

f 1 y . So, f 1 x is

of personalized T-shirts that can be made for a given cost. 105. This situation could be represented by a one-to-one function if the runner does not stop to rest. The inverse function would represent the time in hours for a given number of miles completed. 106. This situation could be represented by a one-to-one function if the population continues to increase. The inverse function would represent the year for a given population.

NOT FOR SALE

Chapter 1

Functions ctions and Their Graphs

Section 1.10 Mathematical Modeling and Variation 1. variation; regression

14. 5

2. sum of square differences

3. least squares regression

3 2

4. correlation coefficient

5. directly proportional

x 1

6. constant of variation

The line appears to pass through 2, 5.5 and 6, 0.5 ,

7. directly proportional

54 x 8.

so its equation is y

8. inverse 15. 9. combined

y 5

10. jointly proportional

11.

Number of people (in thousands)

85,000

80,000

75,000

70,000

Using the points 2, 2 and 4, 1 , y

65,000

12 x 3.

16.

8 10 12 14 16 18 20

Year (8 â&#x2020;&#x201D; 1998)

The model fits the data well.

12. The model is not a good fit for the actual data.

Winning time (in minutes)

5.4 5.2

x 1

4.8 4.6 4.4 4.2 4.0 3.8

The line appears to pass through 0, 2 and 3, 3 so its equation is y

1x 3

t 0

16 24 32 40 48 54

Year (0 â&#x2020;&#x201D; 1950)

13.

y 5 4

2 1 x 1

Using the point 0, 3 and 4, 4 , y

1x 4

NOT FOR SALE Section 1.10

17. (a)

21.

y 240

Length (in feet)

Mathematical Modeling aand Variation y

kx k 10

2050

220 200

205

180

k 205 x

160

Year (20 â&#x2020;&#x201D; 1920)

y 162.3 y

1.02 t 32

1.02t 129.66

Model from part (c): y 1.01 112 130.82

243.94 feet

k 5

1x 5

112

243.9 feet

290 x 3

1 5

24.

Model from part (b): y 1.02 112 129.66

23. y

(d) The models are similar.

k 6

290 3

(b) Using the points 32, 162.3 and 96, 227.7 : 227.7 162.3 m 96 32 | 1.02

580

20 28 36 44 52 60 68 76 84 92 100 108

18. (a) and (c)

22.

140

k 24

25.

18 x

kx k 4

1200

y 0

26.

The model fits the data well.

(d) 2017 o use t

Model from part (b). 40.6 27 204

27. k

k 7x

x y

k 2

1 x

(e) Each year, the gross ticket sales for Broadway shows in New York City increase by about $40.6 million. y

1300

In 2017, the annual gross ticket sales will be about $1300 million.

19.

k S

40.6t 204

(b) S

100

y 100 80 60

20.

k 5

12 5

40 20 x 2

12 x 5

NOT FOR SALE

Chapter 1

28. k

Functions ctions and Their Graphs

31. k

x y

128

200

2, n

x 2 x

1 2

1 3

1 4

1 5

y y

200 160

120

4 5

3 5

2 5 1 5

x 2

1 2

29. k

32. k

x y

1 x3 2

108

256

500

5, n

5 x

500

x y

5 2

5 4

5 6

5 8

1 2

400

5 2

300

200

3 2

100

1 x

1 2

1, 4

30. k

x y

1 x3 4

128

250

33. k

10 x k x2

5 2

5 8

5 18

5 32

1 10

250 y

200 5 2

150

100 3 2

1 x

1 2

x 2

NOT FOR SALE Section 1.10

34. k

40.

x k x2

Mathematical Modeling aand Variation

5 4

5 9

5 16

1 5

k x k 5 k

y 24 120

120 x

y 5

This equation checks with the other points given in the table.

4 3 2 1

41. A

kr 2

42. V

ke3

43. y

k x2

x 2

35. The graph appears to represent y

4 x, so y varies

inversely as x.

36. The graph appears to represent y

3 x, 2

k x k 5 k

1 5

5 x

This equation checks with the other points given in the table.

38. y

2 5

2x 5

7 7 10

45. F

kg r2

46. z

kx 2 y 3

47. R

k T Te

48. P

k V

49. R

kS S L

50. F

km1m2 r2

51. S

4S r 2

The surface area of a sphere varies directly as the square of the radius r.

This equation checks with the other points given in the table.

39.

so y varies

directly with x.

37. y

44. h

52. r

d t

Average speed is directly proportional to the distance and inversely proportional to the time.

kx k 10

53. A

k 7 10 x

1 bh 2

The area of a triangle is jointly proportional to its base and height.

This equation checks with the other points given in the table.

54. V

S r 2h

The volume of a right circular cylinder is jointly proportional to the height and the square of the radius.

55.

k 3

Sr2

7 28

59.

k k 2 3x 2 y

62.

1.5

2 xy

2.16 25.83

63.

113.75

k 11 3

4158

14rs 3

0.035

64.

60.

P 28 3 28 81 3 42 2 27 3 18 P

k 42

92 k k

k 3250

0.035 P

0.0325

65.

211.25

kx y2

k 6500

k 0.0325 P

k 13

33 13

24 287 24 pq 287 s 2

krs 3

1.2

1.5 1.44

4.1 6.3

k 4 8

2x2 3y kpq s2 k 4.1 6.3

kxy

24 36 2 3

28 x

k 6

k x k 4 k

kx 2 y

75 x

58.

61.

k x k 25 k

57.

Functions ctions and Their Graphs

kr 2

A 9S

56.

NOT FOR SALE

Chapter 1

33 x 13

When x

10 inches, y | 25.4 centimeters.

18 x y2

When x

20 inches, y | 50.8 centimeters.

66.

k 14

53 14

53 x 14

5 gallons: y

53 14

| 18.9 liters

INSTRUCTOR USE ONLY 25 gallons: y

53 14

| 94.6 liters

NOT FOR SALE Section 1.10

67.

69.

0.12

k 220

3 5500

3 F 5500

0.16

3 F 5500

880 3

3 5300

T l

440

T l

1.25T 1.2l

440l T

71.

176 23 newtons

where f

1.2 fl 1.25T

and

440l T

1.2 fl 1.25T

440l 1.25T

1.2 fl T

440 1.25T 242,000T 242,000 168,055.56

1.2 f

k 15

1 15

1 F 15

8 2

1F 15

60 lb per spring

Combined lifting force

0.076

| 0.05 meter

3 F 5300

530 3

72.

70. d

3 F 5300

(b) 0.1

k 25 k

F | 39.47.

kF k 265

(a) d

No child over 39.47 pounds should use the toy.

0.15

0.076 F d When the distance compressed is 3 inches, we have 3 0.076 F

d 1.9

The required force is 293 13 newtons.

68.

Mathematical Modeling aand Variation

frequency, T

tension, and l

120 lb

kv 2 2

§1· k¨ ¸ © 4¹ 0.32

0.32v 2

0.12

0.32v 2

0.02

0.12 0.32

3 2 2

3 8 6 | 0.61 mi hr 4

length of string

! 0

T 2

1.44 f T

! 0

1.44 f 2 f2

f | 409.95 vibrations per second

NOT FOR SALE

Chapter 1

73.

Functions ctions and Their Graphs

kmh k 120 1.8

2116.8

2116.8 120 1.8

k W

9.8

k 2 w d 2

(a) load

9.8mh

When m have W

kwd 2 l

74. Load

kwd 2 l

The safe load is unchanged.

100 kilograms and h 1.5 meters, we 9.8 100 1.5 1470 joules.

k 2 w 2d

(b) load

8kwd 2 l

The safe load is eight times as great. k 2 w 2d

4kwd 2 l

The safe load is four times as great. kw d 2

(d) load

1 4 kwd 2 l

The safe load is one-fourth as great.

75. (a) Temperature (in Â°C)

C 5 4 3 2 1

d 2000

4000

Depth (in meters)

(b) Yes, the data appears to be modeled (approximately) by the inverse proportion model. k1 1000 4200 k1 4.2

k2 2000 3800 k2

1.9

k3 3000 4200 k3

1.4

4200 3800 4200 4800 4500 5

k4 4000 4800 k4

1.2

4300, Model: C

k5 5000 4500 k5

0.9

4300 d

6000 0

(e) 3 d

4300 d 4300 3

1 1433 meters 3

NOT FOR SALE

Review Exercises ffor Chapter 1

262.76 x 2.12

76. y (a)

78. False. The closer the value of r is to 1, the better the fit. 79. (a) y will change by a factor of one-fourth.

0.2

(b) y will change by a factor of four.

80. (a) The data shown could be represented by a linear model which would be a good approximation.

55 0

(b) y

(b) The points do not follow a linear pattern. A linear model would be a poor approximation. A quadratic model would be better.

262.76

25 2.12

| 0.2857 microwatts per sq. cm.

77. False. S is a constant, not a variable. So, the area A varies directly as the square of the radius, r.

(d) The data shown could be represented by a linear model which would be a good approximation.

Review Exercises for Chapter 1 1. x 2 6 x 27 0

3 x 9 0

Key numbers: x

3, x

Test intervals: f, 3 , 3, 9 , 9, f

Test: Is x 3 x 9 0? By testing an x-value in each test interval in the inequality, we see that the solution set is 3, 9 . x2 2 x t 3

x2 2 x 3 t 0

x 3 x 1 t 0 Key numbers: x

1, x

Test intervals: f, 1 , 1, 3 , 3, f

Test: Is ( x 3)( x 1) t 0? By testing an x-value in each test interval in the inequality, we see that the solution set is f, 1@ Â&#x2030; >3, f . 6x2 5x 4

â&#x2C6;&#x2019;4

6 x2 5x 4 0

3 x

x â&#x2C6;&#x2019;3

4 2 x 1 0

Key numbers: x

1 2

43 , x

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;1

1 2

Test intervals: f, 43 , 43 ,

1 2

, 12 , f

Test: Is 3x 4 2 x 1 0?

By testing an x-value in each test interval in the inequality, we see that the solution set is 43 ,

1 2

NOT FOR SALE

Chapter 1

Functions ctions and Their Graphs

2 x 2 x t 15

2 x 2 x 15 t 0

2 x

5 x 3 t 0 5 , 2

Key numbers: x

Test intervals: f, 3 , 3,

5 2

, 52 , f

Test: Is (2 x 5)( x 3) t 0? ÂŞ5 Âˇ By testing an x-value in each test interval in the inequality, we see that the solution set is ( f, 3] Â&#x2030; ÂŤ , f Â¸ . ÂŹ2 Âš

x 5 0 3 x

Key numbers: x

5, x

Test intervals: f, 3 , 3, 5 , 5, f

x 5 0? 3 x

Test: Is

By testing an x-value in each test interval in the inequality, we see that the solution set is ( f, 3) Â&#x2030; (5, f).

6. Rational equations, equations involving radicals, and absolute value equations, may have â&#x20AC;&#x153;solutionsâ&#x20AC;? that are extraneous. So checking solutions, in the original equations, is crucial to eliminate these extraneous values. 7. y

9. y

x 2 3x

â&#x20AC;&#x201C;1

â&#x20AC;&#x201C;2

3x 5 y

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;11

â&#x20AC;&#x201C;8

â&#x20AC;&#x201C;5

â&#x20AC;&#x201C;2

5 4

x â&#x20AC;&#x201C;3 â&#x20AC;&#x201C;2 â&#x20AC;&#x201C;1

1 x â&#x20AC;&#x201C;3

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;1

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;3

â&#x20AC;&#x201C;1 â&#x20AC;&#x201C;2 â&#x20AC;&#x201C;3

10. y

2x2 x 9

â&#x20AC;&#x201C;4 â&#x20AC;&#x201C;5

8. y

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;1

â&#x20AC;&#x201C;6

â&#x20AC;&#x201C;9

â&#x20AC;&#x201C;8

â&#x20AC;&#x201C;3

12 x 2 y

â&#x20AC;&#x201C;4

â&#x2C6;&#x2019;5 â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;1

x 1

3 4 5

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;4

10 8 â&#x2C6;&#x2019;9

6 4

INSTRUCTOR S R USE ONLY â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019; 2

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;2

NOT FOR SALE

Review Exercises ffor Chapter 1

11. y

2x 7

13. y

x-intercept: Let y

3 4 2

2x 7

x-intercepts: 0

3 4 Â&#x; x 3

2 0 7

y -intercept: y

9 4

0, 7

x 1 3

14. y

For x 1 0, 0

x 1 3, or 4

0 1 3 or y

3 r2

Â&#x; x

5 or x

3 4 2

x 4 x2

4 x2

4 x

2, 0 , 4, 0

x 1 3

Â&#x; x

x 4 x2

x-intercepts: 0

x 1 3, or 2

0, 5

For x 1 ! 0, 0

y-intercept: Let x

5, 0 , 1, 0

12. x-intercept: Let y

Â&#x; x 3

72 , 0

y -intercept: Let x

0, 0 , 2, 0 , 2, 0

0. y -intercept: y

0Â&#x2DC;

4 0

0, 0

0, 2

15. y

4x 1

Intercepts:

14 , 0 , 0, 1

4 x 1 Â&#x; y

4x 1 Â&#x; y

4 x 1 Â&#x; y

16. y

4 x 1 Â&#x; No y -axis symmetry 4 x 1 Â&#x; No x-axis symmetry

4 x 1 Â&#x; No origin symmetry

5x 6

Intercepts:

56 , 0 , 0, 6

5 x 6 Â&#x; y

5x 6 Â&#x; y

5 x 6 Â&#x; y

5 x 6 Â&#x; No y -axis symmetry 5 x 6 Â&#x; No x-axis symmetry

5 x 6 Â&#x; No origin symmetry

NOT FOR SALE

Chapter 1

17. y

Functions ctions and Their Graphs

5 x2

5, 0 , 0, 5

Intercepts: r y

5 x Â&#x; y

5 x2 Â&#x; y

5 x Â&#x; y

5 x 2 Â&#x; y -axis symmetry

4 3

5 x 2 Â&#x; No x-axis symmetry

2 1

5 x 2 Â&#x; No origin symmetry

â&#x2C6;&#x2019; 4 â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1

x 1

â&#x2C6;&#x2019;2

18. y

x 2 10

Intercepts: r 10, 0 , 0, 10

â&#x2C6;&#x2019; 6 â&#x2C6;&#x2019;4

y-axis symmetry

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;6

â&#x2C6;&#x2019; 12

19. y

x3 3

Intercepts: 3 3, 0 , 0, 3

x 3

3 Â&#x; y

x3 3 Â&#x; y

x 3

20. y

x3 3 Â&#x; No y -axis symmetry x3 3 Â&#x; No x-axis symmetry

3 Â&#x; y

x3 3 Â&#x; No origin symmetry

6 x3

Intercepts:

6, 0 , 0, 6

6 x Â&#x; y

6 x3 Â&#x; y

6 x Â&#x; y

21. y

6 x Â&#x; No y -axis symmetry 6 x3 Â&#x; No x-axis symmetry

6 x3 Â&#x; No origin symmetry

x 5

Domain: [ 5, f)

Intercepts: 5, 0 , 0, y

6 5 4

x 5 Â&#x; No y -axis symmetry

x 5 Â&#x; y

x 5 Â&#x; No x-axis symmetry x 5 Â&#x; No origin symmetry

1 â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;5 â&#x2C6;&#x2019; 4 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1

x 1

NOT FOR SALE

Review Exercises ffor Chapter 1

22. y

x 9

Intercept: 0, 9

x 9 Â&#x; y

23. x 2 y 2

x 9 Â&#x; y -axis symmetry x 9 Â&#x; No x-axis symmetry x 9 Â&#x; No origin symmetry

25.

2 y 2

y 0

Center: 0, 0

x 2

Radius: 3

Center: 2, 0

Radius: 4

24. x 2 y 2

Center: 0, 0

Radius: 2

26. x 2 y 8

Center: 0, 8

Radius: 9

100

NOT FOR SALE

Chapter 1

Functions ctions and Their Graphs

27. Endpoints of a diameter: 0, 0 and 4, 6

Â§ 0 4 0 6 Â· , Center: Â¨ Â¸ 2 Â© 2 Â¹

Radius: r

2, 3

0 3 0

4 9

Standard form: x 2 y 3

x 28. F

(a)

5 x, 4

2 y 3

13 2

0 d x d 20

31. y

Slope: m x

y-intercept: 0, 6

(b)

32. x

Slope: m is undefined. y-intercept: none

10, F

50 4

12.5 pounds.

3 x 13

Slope: m

33. 6, 4 , 3, 4

y-intercept: 0, 13

4 4

6 3

4 4 6 3

8 9

3 â&#x2C6;&#x2019; 9 â&#x2C6;&#x2019;6 â&#x2C6;&#x2019; 3

x 3

â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;6

30. y

10 x 9

Slope: m

y-intercept: 0, 9

12 9

34. 3, 2 , 8, 2

6 3 â&#x2C6;&#x2019;9

â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;3

3 â&#x2C6;&#x2019;3

2 2 3 8

0 11

NOT FOR SALE

Review Exercises ffor Chapter 1

35. 10, 3 , m y 3

39. Point: 3, 2

12 12 x 10

5x 4 y 5 x 4

8 2

y 3

12 x 5

12 x 2

(a) Parallel slope: m y 2

5 4

y 2

5 x 4

15 4

5 x 4

23 4

(b) Perpendicular slope: m y 2

36. 8, 5 ,

y 2

54 x

12 5

54 x

2 5

40. Point: 8, 3 , 2 x 3 y

2 0

6 1

5 3

2x 3

y y

1 2

6 11

y 2

5 y 10 5y y

23 x 8

3y 9

2 x 16

3y y

2x 7 23 x

7 3

(b) Perpendicular slope: m y 3

3 2

3 x 24

3 x 30 3 x 2

3 2

2y 6 y

38. 11, 2 , 6, 1

y 3

2 7

2 x 1

7 2 x 1

7 2 2 x 7 7

y 0

5 2x

(a) Parallel slope: m

37. 1, 0 , 6, 2

54 x 3

0 x 8

y 5

101

41. Verbal Model : Sale price = List price Discount

1 5

1 x 11

5 x 11

Labels :

List price = L Discount = 20% of L = 0.2L

x 1

Sale price = S

Equation :

S = L 0.2L S = 0.8 L

1 1 x 5 5

102

NOT FOR SALE

Chapter 1

Functions ctions and Their Graphs

42. Verbal Model : Hourly wage = Base wage per hour + Piecework rate Â&#x2DC; Number of units

Labels :

Hourly wage = W Base wage = 12.25 Piecework rate = 0.75 Number of units = x

Equation :

12.25 0.75 x

43. 16 x y 4

49. f x

16 x

25 x 2 t 0

Domain:

5 x 5 x

No, y is not a function of x. Some x-values correspond to two y-values. 44. 2 x y 3

2x 3

Critical numbers: x

t 0

Test intervals: f, 5 , 5, 5 , 5, f

Test: Is 25 x 2 t 0? Solution set: 5 d x d 5

Yes, the equation represents y as a function of x. 45. y

25 x 2

1 x

Domain: all real numbers x such that 5 d x d 5, or > 5, 5@

Yes, the equation represents y as a function of x. Each x-value, x d 1, corresponds to only one y-value. 46.

y y

x 2 corresponds to y

x 2 or

x 2.

No, y is not a function of x. Some x-values correspond to two y-values. 47. f x

x2 1

(a) f 2

2 2

(b) f 4

4 2

t 2

(d) f t 1

x x2 x 6 x x 2

x 3

50. h x

5 17 t4 1

Domain: All real numbers x except x

2, 3

1 1 2

t 2t 2 2

48. h x

Â2 x 1, x d 1 ÂŽ 2 ÂŻx 2, x ! 1

(a) h 2

2 2 1

(b) h 1

2 1 1

3 1

02 2

51. v t

(d) h 2

2 2

v 1

52. 0 t

32t 48 16 feet per second 32t 48 48 32

1.5 seconds

NOT FOR SALE

Review Exercises ffor Chapter 1

53. f x

103

2 x 2 3x 1

ÂŞ2 x h 2 3 x h 1Âş 2 x 2 3 x 1

ÂŹ Âź h

f x h f x

2 x 2 4 xh 2h 2 3 x 3h 1 2 x 2 3 x 1 h h 4 x 2h 3

h 4 x 2h 3, h z 0 f x

54.

x3 5 x 2 x

f x h

h 5 x h x h

x3 3x 2 h 3 xh 2 h3 5 x 2 10 xh 5h 2 x h f x h f x

x3 3x 2 h 3 xh 2 h3 5 x 2 10 xh 5h 2 x h x3 5 x 2 x h

3 x 2 h 3 xh 2 h3 10 xh 5h 2 h h h 3 x 2 3 xh h 2 10 x 5h 1

h 3 x 2 3 xh h 2 10 x 5h 1, h z 0

55. y

60. f x

A vertical line intersects the graph no more than once, so y is a function of x. 4 y

56. x

57. f x

3x 2 16 x 21

3 x

7 x 3

3x 7 x 58. f x

0 or 7 3

x x 1

61. f x

0 or

x 1

x x 1

f is increasing on 0, f . f is decreasing on f, 1 .

x 3

f is constant on 1, 0 .

5x2 4 x 1

5 x

x3 x 2

A vertical line intersects the graph more than once, so y is not a function of x.

x3 x 2

1 x 1

0 0

5x 1

0 Â&#x; x

1 5

x 1

0 Â&#x; x

62. f x

8x 3 11 x

59. f x

8x 3 11 x 8x 3

0 0 83

f is increasing on 2, 0 and 2, f .

INSTRUCTOR USE ONLY x

f is decreasing on f, 2 and 0,, 2 .

104

Chapter 1

63. f x

NOT FOR SALE

Functions ctions and Their Graphs

69. (a) f 2

x2 2 x 1

3 6

y 6

x3 4 x 2 1

y 6

Relative maximum: 0, 1

12 4

4 0

0 to x2

f x

70. (a) f 0

8 2 2

7 3

2 2.

f x

x 4 20 x 2

2 3 to x2

y 5

f x

20 x

x 20 x 4

4 0

The average rate of change of f from x1

5, f 4

0, 5 , 4, 8

8 5

4 2 2 2 4

f x

3x 6

x 1

f 7 f 3

67.

3 x 2

(b)

The average rate of change of f from x1 is 4.

is 1

x2 8x 4

f 4 f 0

66. f x

9 3

1 2

Relative minimum: 2.67, 10.48

65. f x

Points: 2, 6 , 1, 3

Relative maximum: 1, 2

64. f x

6, f 1

f x

3 4

3 x 0

4 3 x 5 4 3 x 5 4

(b)

The function is even. 68.

f x

x2 3

2 x

x2 3

f x

The function is odd.

71. f x

3 x2 y

6 4

â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;6

NOT FOR SALE

Review Exercises ffor Chapter 1

72. f x

x 1

76. (a) f x

(b) h x

105

2 2 3

Horizontal shift 2 units to the right; vertical shift 2 units upward (c)

73. g x

1 x 5

(d) h x

f x 2 2

77. (a) f x

(b) h x

x 4

Vertical shift 4 units upward, reflection in the x-axis (c) 74. f x

x t 1 Â5 x 3, ÂŽ 1 4 x 5, x ÂŻ

(d) h x

78. (a) f x

75. (a) f x

(b) h x

f x 4 x x 3 5

Horizontal shift 3 units to the left; vertical shift 5 units downward

x2 9

Vertical shift 9 units downward

(c)

(d) h x

f x 9

(d) h x

f x 3 5

106

NOT FOR SALE

Chapter 1

79. (a) f x

(b) h x

Functions ctions and Their Graphs

82. (a) f x

(b) h x

x 2 3 2

x 1 9

Reflection in the x-axis, a horizontal shift 1 unit to the left, and a vertical shift 9 units upward

Horizontal shift two units to the left, vertical shift 3 units upward, reflection in the x-axis. (c)

(c)

(d) h x

80. (a) f x

(b) h x

(d) h x

f x 2 3

83. (a) f x

x2 1 2

(b) h x

1 2 2

Horizontal shift one unit to the right, vertical shrink, vertical shift 2 units downward (c)

(d) h x

81. (a) f x

(b) h x

1 2

(c)

84. (a) f x

axb a xb 6

(b) h x

(d) h x

5 f x 9

13 x3

Reflection in the x-axis; vertical shrink each y-value is multiplied by 13

Reflection in the x-axis and a vertical shift 6 units upward (c)

axb 5a x 9b

Horizontal shift 9 units to the right and a vertical stretch (each y-value is multiplied by 5)

(d) h x

f x 1 2

f x 1 9

(c)

f x 6

(d) h x

13 f x

NOT FOR SALE

Review Exercises ffor Chapter 1

85. f x

(a) (b) (c)

x 2 3, g x

87. f x

2x 1

x2 3 2 x 1 x2 2 x 2 f g x x 2 3 2 x 1 x 2 2 x 4 fg x x 2 3 2 x 1 2 x3 x 2 6 x 3 f

g x

Â§fÂˇ (d) Â¨ Â¸ x

ÂŠgÂš

86. f x

3, g x

(a)

D g x

1 3

3 x

g x

f x g x

x 4

3 x

(b)

g x

f x g x

x2 4

3 x

(c)

fg x

f x

Â§fÂˇ (d) Â¨ Â¸ x

ÂŠgÂš

3 x

(b)

D f x

1 3

8 3

g f x

13 x 3

3 13 x 3 1 g

x 9 1

x2 4 , Domain: x 3 3 x

g x

3 x

Domain: all real numbers

(a)

f x g x

f g x

f 3 x 1

x 3 1 , Domain: x z 2x 1 2

x 4, g x

3x 1

The domains of f and g are all real numbers.

88. f x

1x 3

107

x 8

Domain: all real numbers

x3 4, g x

x 7

The domains of f x and g x are all real numbers. (a)

D g x

f g x

x 7

x 7 4 x 3 Domain: all real numbers (b)

D f x

g f x

x3 4 7

x3 3

Domain: all real numbers 89. N T t

25 2t 1 50 2t 1 300, 2 d t d 20 2

25 4t 2 4t 1 100t 50 300 100t 2 100t 25 100t 250 100t 2 275 The composition N T t

represents the number of bacteria in the food as a function of time. 90. When N

750

750, 100t 2 275

100t

475

4.75 2.18 hours.

After about 2.18 hours, the bacterial count will reach 750.

108 91.

Chapter 1 f x

NOT FOR SALE

Functions ctions and Their Graphs

3x 8

3x 8 3y 8 3y x 8 y 3 1 y x 8

3 1 So, f 1 x

x 8

y x x 8

f f 1 x

Â§1 Âˇ f Â¨ x 8 Â¸ ÂŠ3 Âš

f 1 f x

f 1 3 x 8

92. f x

y x 5x y

x 4 5 x 4 5 y 4 5 y 4 5x 4

So, f 1 x

5x 4

5x 4 4 5x x 5 5 Â§ x 4Âˇ Â§ x 4Âˇ f 1 Â¨ x 4 4 Â¸ 5Â¨ Â¸ 4 ÂŠ 5 Âš ÂŠ 5 Âš

f f 1 x

f 5 x 4

f 1 f x

93. f x

Â§1 Âˇ 3Â¨ x 8 Â¸ 8 x 8 8 ÂŠ3 Âš 1 1 3x 8 8

3 x x 3 3

x f x

1x 2

3 (b)

1x 2

1y 2

x 3

1y 2

95. (a)

No, the function does not have an inverse because some horizontal lines intersect the graph twice.

2 x 3

94. h t

y 2x 6

2 t 3

Yes, the function has an inverse because no horizontal lines intersect the graph at more than one point. The function has an inverse.

(d) The domains and ranges of f and f 1 are the set of all real numbers.

NOT FOR SALE

Review Exercises ffor Chapter 1

f x

x 1

y 1

99.

y 1

x2 x 1

f 1 x

x 2 1, x t 0

Value of shipments (in billions of dollars)

96. (a)

109

14 12 10 8 6 4 2 t 2

10 12

Year (4 â&#x2020;&#x201D; 2004)

Note: The inverse must have a restricted domain.

The model fits the data well.

(b)

100. T

3 k

k r k 65 3 65

195

195 r 80 mph, When r T

is > 1, f .

The range of f and the domain of f 1 is >0, f . 101. 97. f x

2 x 4 is increasing on 4, f .

0.05

2 x 4

0.05 14 8 2

2 y 4 , x ! 0, y ! 4

x 2

y 4

98. f x

2 2

Let f x

$44.80

102. False. The graph is reflected in the x-axis, shifted 9 units to the left, then shifted 13 units downward.

y x 4, x ! 0 2 x 2 is increasing on 2, f . x 2, x ! 2, y ! 0.

x 2

y 2, x ! 0, y ! 2

k 16 6

f 1 x

x 2

28.80

khw2

2 x 4 , x ! 4 and y ! 0.

x 4 2

Let f x

195 2.4375 hours 80 | 2 hours, 26 minutes.

103. True. If f x

x3 and g x

x , then the domain

of g is all real numbers, which is equal to the range of f and vice versa.

y, x ! 0, y ! 2 x 2, x ! 0

110

Chapter 1

NOT FOR SALE

Functions ctions and Their Graphs

Problem Solving for Chapter 1 1. (a) W1

0.07 S 2000

(b) W2

0.05S 2300

(c)

Point of intersection: 15,000, 3050

Both jobs pay the same, $3050, if you sell $15,000 per month. (d) No. If you think you can sell $20,000 per month, keep your current job with the higher commission rate. For sales over $15,000 it pays more than the other job. 2. Mapping numbers onto letters is not a function. Each number between 2 and 9 is mapped to more than one letter.

^ 2, A , 2, B , 2, C , 3, D , 3, E , 3, F , 4, G , 4, H , 4, I , 5, J , 5, K , 5, L , 6, M , 6, N , 6, O , 7, P , 7, Q , 7, R , 7, S , 8, T , 8, U , 8, V , 9, W , 9, X , 9, Y , 9, Z ` Mapping letters onto numbers is a function. Each letter is only mapped to one number.

^ A, 2 , B, 2 , C , 2 , D, 3 , E , 3 , F , 3 , G, 4 , H , 4 , I , 4 , J , 5 , K , 5 , L, 5 , M , 6 , N , 6 , O, 6 , P, 7 , Q, 7 , R, 7 , S , 7 , T , 8 , U , 8 , V , 8 , W , 9 , X , 9 , Y , 9 , Z , 9 ` 3. (a) Let f x and g x be two even functions.

Then define h x

h x

f x r g x .

f x r g x

Then define h x

h x

f x r g x .

f x r g x

f x r g x because f and g are even

f x r g x because f and g are odd

h x

So, h x is also odd. If f x z g x

So, h x is also even. (c) Let f x be odd and g x be even. Then define h x

h x

(b) Let f x and g x be two odd functions.

f x r g x .

f x r g x

f x r g x because f is odd and g is even z h x

z h x

So, h x is neither odd nor even.

NOT FOR SALE

Problem Solving ffor Chapter 1

4. f x

g x

D f x

x and g D g x

111

These are the only two linear functions that are their own inverse functions since m has to equal 1 m for this to be true. x c

General formula: y 5.

f x

a2 n x 2 n a2 n 2 x 2 n 2 " a2 x 2 a0 a2 n x

a2 n 2 x

2n 2

" a2 x a0 2

a2 n x 2 n a2 n 2 x 2 n 2 " a 2 x 2 a0

f x

So, f x is even. 6. It appears, from the drawing, that the triangles are equal; thus x, y 6, 8 .

The line between 2.5, 2 and 6, 8 is y

12 x 7

The line between 9.5, 2 and 6, 8 is y

12 x 7

7. (a) April 11: 10 hours April 12: 24 hours April 13: 24 hours

16 . 7

2 April 14: 23 hours 3

128 . 7

The path of the ball is:

f x

Total:

ÂÂ°12 x 16 , 2.5 d x d 6 7 7 ÂŽ 12 128 Â°Ě&#x201E; 7 x 7 , 6 x d 9.5

(b) Speed

2 81 hours 3

distance time

2100 2 81 3

180 7

5 25 mph 7

180 t 3400 7 1190 Domain: 0 d t d 9 Range: 0 d D d 3400

(d)

112

Chapter 1

8. (a)

(b) (c) (d) (e)

NOT FOR SALE

Functions ctions and Their Graphs

f x2 f x1

f 2 f 1

x2 x1

1 0 1

2 1

f x2 f x1

f 1.5 f 1

x2 x1

1.5 1

0.75 0 0.5

f x2 f x1

f 1.25 f 1

x2 x1

1.25 1

f x2 f x1

f 1.125 f 1

x2 x1

1.125 1

1.5

0.4375 0 0.25

1.75

0.234375 0 0.125

f x2 f x1

f 1.0625 f 1

x2 x1

1.0625 1

1.875

0.12109375 0 0.625

1.9375

(f ) Yes, the average rate of change appears to be approaching 2. (g) a. 1, 0 , 2, 1 , m

x 1

1, y

0.75 0.5

b. 1, 0 , 1.5, 0.75 , m

0.4375 0.25

c. 1, 0 , 1.25, 0.4375 , m

9. (a)â&#x20AC;&#x201C;(d) Use f x

(a)

D g x

(b)

D g

g 1 x

g 1 D

4 x and g x

f x 6

x 24 4

1.9375, y

1.9375 x 1.9375

2x 2

x 6.

4 x 6

1.875 x 1.875

4 x 24

the length of the trip over land is

Â§1 Âˇ g 1 Â¨ x Â¸ ÂŠ4 Âš

(e) f x

x 1 and g x

D g x

D g

f 2 x

f 1 x

g 1 x

1 x 2

g 1 D

f 1 x

4 x2 2

T x

1 3 x

1 4 x 4 2

4 x 6 x 10. 2

(c)

2x 1

1 2

8 x3 1

x 1 (d) T x is a minimum when x

x 1

g 1

(b) Domain of T x : 0 d x d 3

1 x 6 4

2 x 3

x 1 8

1 3 x .

The total time is

1 x 6 4

1 2

f 1 x

22 x 2 , and

10. (a) The length of the trip in the water is

1 x 4 x 6

(d)

2 x 1 , y

o 2, y

1.875, y

0.12109375 0.0625

e. 1, 0 , 1.0625, 0.12109375 , m

1, 0 , m

1.75 x 1.75

1.75, y

0.234375 0.125

d. 1, 0 , 1.125, 0.234375 , m

(h) 1, f 1

1.5 x 1.5

1.5, y

x 1

1 2

x 1

(e) Answers will vary. Sample answer: To reach point Q in the shortest amount of time, you should row to a point one mile down the coast, and then walk the rest of the way.

(f ) Answers will vary. (g) Conjecture: f D g

g 1 D

f 1 x

NOT FOR SALE

Problem Solving ffor Chapter 1

11. H x

113

Â1, x t 0 Â® Â¯0, x 0

(a) H x 2

(b)

H x 2

(c)

H x

(d) H x

(e)

1 H 2

(f )

H x 2 2

3 2

â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;1

x 1

â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;3

12. f x

1 1 x

(a) Domain: all real numbers x except x Range: all real numbers y except y (b) f f x

1 Â§ x 1Â· 1 Â¨ Â¸ Â© x Â¹

1 1 x

The graph is not a line. It has holes at 0, 0 and

1, 1 .

Â§ 1 Â· fÂ¨ Â¸ Â©1 x Â¹ 1 1 1 x 1 Â§ 1 Â· 1 Â¨ Â¸ 1 x Â©1 x Â¹ x 1 1 x x x

Domain: all real numbers x except x x 1

Â§ x 1Â· fÂ¨ Â¸ Â© x Â¹

0 and

114

13.

Chapter 1

f D g D h

f D g D h x

14. (a) f x 1

NOT FOR SALE

Functions ctions and Their Graphs f g D h x

D g h x

f g h x

D g D h x

(b) f x 1

(d) f x

(e) f x

(f )

(g) f x

f x

NOT FOR SALE

Problem Solving ffor Chapter 1

15.

f x

f 1 x

â&#x20AC;&#x201C;4

â&#x20AC;&#x201D;

â&#x20AC;&#x201C;3

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;1

â&#x20AC;&#x201D;

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;1

â&#x20AC;&#x201C;3

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;4

â&#x20AC;&#x201D;

â&#x20AC;&#x201C;3

(a)

(c)

f f 1 x

â&#x20AC;&#x201C;4

f f 1 4

f 2

â&#x20AC;&#x201C;2

f f 1 2

f 0

f f

f 1

f f 1 4

f 3

â&#x20AC;&#x201C;3

f 3 f 1 3

4 1

â&#x20AC;&#x201C;2

f 2 f 1 2

f 0 f 1 0

f 1 f 1 1

â&#x20AC;&#x201C;3

f 3 f 1 3

4 1

â&#x20AC;&#x201C;2

f 2 f 1 2

1 0

f 0 f

f 1 f 1 1

(b)

(d)

Â&#x2DC; f 1 x

f 1 x

2 1

3 2

f 1 x

â&#x20AC;&#x201C;4

f 1 4

1 0

â&#x20AC;&#x201C;3

f 1 3

2 1

f 1 0

f 1 4

3 2

115

116

Chapter 1

NOT FOR SALE

Functions ctions and Their Graphs

Practice Test for Chapter 1 1. Given the points 3, 4 and 5, 6 , find (a) the midpoint of the line segment joining the points,

and (b) the distance between the points. 7 x.

2. Graph y

3. Write the standard equation of the circle with center 3, 5 and radius 6. 4. Find the equation of the line through 2, 4 and 3, 1 . 5. Find the equation of the line with slope m

4 3 and y-intercept b

6. Find the equation of the line through 4, 1 perpendicular to the line 2 x 3 y

7. If it costs a company $32 to produce 5 units of a product and $44 to produce 9 units, how much does it cost to produce 20 units? (Assume that the cost function is linear.) 8. Given f x

x 2 2 x 1, find f x 3 .

9. Given f x

4 x 11, find

f x f 3

x 3

10. Find the domain and range of f x

36 x 2 .

11. Which equations determine y as a function of x?

(a) 6 x 5 y 4 (b) x y 2

x2 6

12. Sketch the graph of f x

x 2 5.

13. Sketch the graph of f x

x 3.

14. Sketch the graph of f x

Â2 x 1, if x t 0, ÂŽ 2 ÂŻx x, if x 0.

15. Use the graph of f x

x to graph the following:

(a) f x 2

(b) f x 2 16. Given f x

(a)

(b)

fg x

3x 7 and g x

2 x 2 5, find the following:

f x

17. Given f x

x 2 2 x 16 and g x

18. Given f x

x 3 7, find f 1 x .

2 x 3, find f g x

NOT FOR SALE

Practice Test ffor Chapter 1

117

19. Which of the following functions have inverses?

(a) f x

x 6

(b) f x

ax b, a z 0

x 3 19

3 x , 0 x d 3, find f 1 x . x

20. Given f x

Exercises 21â&#x20AC;&#x201C;23, true or false? 21. y 22.

3 x 7 and y

D g

1x 3

4 are perpendicular.

g 1 D f 1

23. If a function has an inverse, then it must pass both the Vertical Line Test and the Horizontal Line Test. 24. If z varies directly as the cube of x and inversely as the square root of y, and z when x 1 and y 25, find z in terms of x and y.

25. Use your calculator to find the least square regression line for the data.

â&#x20AC;&#x201C;2

2.4

3.1

4.7

NOT FOR SALE C H A P T E R 2 Polynomial and Rational Functions Section 2.1

Quadratic Functions and Models .......................................................119

Section 2.2

Polynomial Functions of Higher Degree...........................................134

Section 2.3

Polynomial and Synthetic Division ...................................................151

Section 2.4

Complex Numbers..............................................................................165

Section 2.5

Zeros of Polynomial Functions..........................................................171

Section 2.6

Rational Functions..............................................................................188

Section 2.7

Nonlinear Inequalities ........................................................................202

Review Exercises ........................................................................................................219 Problem Solving .........................................................................................................230 Practice Test .............................................................................................................234

NOT FOR SALE C H A P T E R 2 Polynomial and Rational Functions Section 2.1 Quadratic Functions and Models 8. f x

1. polynomial

4 opens upward and has vertex 4, 0 . 2

Matches graph (c).

2. nonnegative integer; real

9. f x

3. quadratic; parabola

x 2 2 opens upward and has vertex 0, 2 .

Matches graph (b).

4. axis

10. f x

5. positive; minimum

1, 2 .

6. negative; maximum 7. f x

11. f x

2 opens upward and has vertex 2, 0 . 2

4, 0 .

(b)

18 x 2

â&#x2C6;&#x2019;1

3 2 x 2

(d)

3 x 2

â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;4

x 4

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;1

x 1

Vertical stretch

x2 1

(c)

Vertical stretch and reflection in the x-axis

x2 3

(d)

x2 3

1 x

â&#x2C6;&#x2019;2

x 1

â&#x2C6;&#x2019;1

Vertical shift one unit upward

â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;3

Vertical shrink and reflection in the x-axis y

â&#x2C6;&#x2019;6

(b)

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;4

x 1 â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;1

Matches graph (d).

x2 1

â&#x2C6;&#x2019;2

(c)

â&#x2C6;&#x2019;3

Vertical shrink 14. (a) y

x 2 4 opens

1 â&#x2C6;&#x2019;2

Matches graph (a).

x 4 opens downward and has vertex

â&#x2C6;&#x2019;3

4 x 2

12. f x

1 x2 2

1 2 opens upward and has vertex

downward and has vertex 2, 4 . Matches graph (f).

Matches graph (e).

13. (a) y

â&#x2C6;&#x2019;2

Vertical shift one unit downward

â&#x2C6;&#x2019;6

3 â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;2

x â&#x20AC;&#x201C;4

x 2

â&#x2C6;&#x2019;2

Vertical shift three units upward

â&#x2C6;&#x2019;4

Vertical shift three units downward

119

120

Chapter 2

15. (a) y

Polynomial ynomial and Rational Function Functions

(b)

3 x 2

(c)

13 x

(d)

4 2

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;6

x 1

â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;1

Horizontal shift one unit to the right

16. (a) y

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;2

x 1

12 x 2 1 2

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;8

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;1

Horizontal shrink and a vertical shift one unit upward

x 2

Horizontal stretch and a vertical shift three units downward (c) y

â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;2

x 2

â&#x2C6;&#x2019;2

Horizontal shift three units to the left

12 x 2 1 2

â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;8 â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;4

8 10

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;8

Horizontal shift two units to the right, vertical shrink

Horizontal shift two units to the left, vertical shrink

each y-value is multiplied by , reflection in the

each y-value is multiplied by 12 , reflection in

x-axis, and vertical shift one unit upward

x-axis, and vertical shift one unit downward

1 2

(b) y

Âª 1 x 1 Âº 3 Â¬2 Â¼ 2

(d) y

Â¬Âª2 x 1 ÂºÂ¼ 4 2

10 8 6

3 2

â&#x2C6;&#x2019;8 â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019; 2 â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;6

x 1

Horizontal shift one unit to the left, horizontal shrink

Horizontal shift one unit to the right, horizontal stretch each x-value is multiplied by 2 , and vertical

each x-value is multiplied by 12 , and vertical shift

shift three units downward

four units upward

NOT FOR SALE Section 2.1

17. f x

Quadratic Function Functions and Models

21. f x

x2 6x

x 2 8 x 13

x 2 6 x 9 9

x 2 8 x 16 16 13

x 4 2 4, 3

3 9 2

Vertex: 3, 9

Vertex:

Axis of symmetry: x

x 6x

x x 6

x 6

Find x-intercepts:

10 8

x 2 8 x 13 x

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;8 â&#x2C6;&#x2019;10

x-intercepts: 0, 0 , 6, 0

x 2 8 x 16

22. f x

4 16 2

â&#x2C6;&#x2019;2

x x 8

â&#x2C6;&#x2019;10

0 Â&#x; x

â&#x2C6;&#x2019;12

x 12 x 2

x 2 12 x 36

â&#x2C6;&#x2019;16

x y

x 4

x-intercept: 4, 0

23. f x

â&#x2C6;&#x2019;4

0 y

44 44 36

6 4

6 r 2 2i

2 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;2

x 2

8 10 12 14

x 2 14 x 54

x 7 2 7, 5

Vertex:

Axis of symmetry: x Find x-intercepts:

x 2 14 x 54

y 6

x 14 x

x 2 14 x 49

x 7

x 1

3, 0

x 2 14 x 49 49 54

Find x-intercepts: 1

â&#x2C6;&#x2019;3

No x-intercepts

Vertex: 1, 0

x 1

â&#x2C6;&#x2019;2

Not a real number

x 6

Axis of symmetry: x

Vertex:

x 2 12 x 44

â&#x2C6;&#x2019;14

x2 2 x 1

â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1

Axis of symmetry: x Find x-intercepts:

â&#x2C6;&#x2019;8

Find x-intercepts:

20. g x

â&#x2C6;&#x2019;6

Vertex: 4, 0

â&#x2C6;&#x2019;4

x-intercepts: 0, 0 , 8, 0

â&#x2C6;&#x2019;2

Axis of symmetry: x

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019; 3

4 r

x 6 2 6, 12

x2 8x

x 8 x 16

â&#x2C6;&#x2019;7 â&#x2C6;&#x2019;6

x 2 12 x 36 36 44

Find x-intercepts:

19. h x

x 2 12 x 44

Vertex: 4, 16

x-intercepts: 4 r

x 2 8 x 16 16

x 8

16 13

x 8x

Axis of symmetry: x

x 4

x 18. g x

x2 8x

0 Â&#x; x

Find x-intercepts: 2

Axis of symmetry: x

121

x 7 x x

54 54 49 5 r 7r

15 12 9

5 5i

6 3

INSTRUCTOR USE ONLY p 1, 0

x-intercept:

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;1

Not a real number

No xx-intercepts -intercepts intercepts

122

Chapter 2

24. h x

NOT FOR SALE

Polynomial ynomial and Rational Function Functions

x 2 16 x 17

27. f x

x 2 16 x 64 64 17 x

8 81 2

Vertex: 8, 81

1Â· Â§ Â¨x Â¸ 1 2Â¹ Â©

Axis of symmetry: x x 2 16 x 17

17 x 1

0 Â&#x; x

x 1

0 Â&#x; x

x2 x

5 4

28. f x

â&#x2C6;&#x2019;70 â&#x2C6;&#x2019;80

x 2 34 x 289 17

x 17 x

x-intercept: 17, 0

â&#x2C6;&#x2019;25 â&#x2C6;&#x2019;20 â&#x2C6;&#x2019;15 â&#x2C6;&#x2019;10

x 2 3x

â&#x2C6;&#x2019;5

1 4

0 3 r

3 r 2

Â§ 3 x-intercepts: Â¨ Â© 2

Vertex: 15, 0

9 1 2

x 2 30 x 225 15

3 2

Find x-intercepts:

Axis of symmetry: x

x 2 34 x 289

1 4 9Â· 9 1 Â§ 2 Â¨ x 3x Â¸ 4 4 4 Â© Â¹ x 2 3x

Â§ 3 Â· Vertex: Â¨ , 2 Â¸ Â© 2 Â¹

Find x-intercepts:

26. f x

1 5 2

Axis of symmetry: x

3Â· Â§ Â¨x Â¸ 2 2Â¹ Â©

Vertex: 17, 0

No x-intercepts

â&#x2C6;&#x2019;60

Not a real number

â&#x2C6;&#x2019;20

25. f x

â&#x2C6;&#x2019;1

x 2

â&#x2C6;&#x2019;10

1 â&#x2C6;&#x2019;2

10 â&#x2C6;&#x2019;4

1 2

Find x-intercepts:

x-intercepts: 17, 0 , 1, 0

â&#x2C6;&#x2019;8

5 4

Axis of symmetry: x

x 17

â&#x2C6;&#x2019;12

Â§1 Â· Vertex: Â¨ , 1Â¸ Â©2 Â¹

Find x-intercepts:

5 4 1Â· 1 5 Â§ 2 Â¨x x Â¸ 4Â¹ 4 4 Â© x2 x

Â· Â§ 3 2, 0 Â¸, Â¨ Â¹ Â© 2

Â· 2, 0 Â¸ Â¹

Axis of symmetry: x

Find x-intercepts:

x 2 30 x 225

x 15 x

x-intercept: 15, 0

2 1 â&#x2C6;&#x2019; 5 â&#x2C6;&#x2019; 4 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;1

x 1

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;3 x

NOT FOR SALE Section 2.1

29. f x

Quadratic Functions Function and Models

31. h x

x2 2 x 5

4 x 2 4 x 21

x 2 2 x 1 1 5

1Â· Â§ Â§1Â· 4Â¨ x 2 x Â¸ 4Â¨ Â¸ 21 4Â¹ Â© Â© 4Â¹

x 1 6 2

1Â· Â§ 4Â¨ x Â¸ 20 2Â¹ Â©

Vertex: 1, 6

Axis of symmetry: x

Â§1 Â· Vertex: Â¨ , 20 Â¸ Â©2 Â¹

Find x-intercepts:

x2 2 x 5

x 2x 5

Axis of symmetry: x

4 x 2 4 x 21

6, 0 , 1

1 2

Find x-intercepts:

4 20 2

x-intercepts: 1

123

6, 0

0 4r

16 336 2 4

Not a real number

No x-intercepts

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;4

30. f x

x 4 x 1

x 4x 1 2

x 4 x 4 4 1

â&#x2C6;&#x2019;8

â&#x2C6;&#x2019;4

x 2 5 2

Find x-intercepts: x 2 4 x 1

x 4x 1

1Â· Â§ Â§1Â· 2Â¨ x Â¸ 2Â¨ Â¸ 1 4Â¹ Â© Â© 16 Â¹

5, 0 , 2

4 r

5, 0

2 r x-intercepts: 2

2 x2 x 1 1 Â· Â§ 2Â¨ x 2 x Â¸ 1 2 Â¹ Â©

Vertex: 2, 5

Axis of symmetry: x

32. f x

16 4 2 5

1Â· 7 Â§ 2Â¨ x Â¸ 4Â¹ 8 Â©

Â§1 Vertex: Â¨ , Â©4

6 5

7Â· Â¸ 8Â¹

4 3

Axis of symmetry: x

1 4

2 x2 x 1

4 2

1 â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;5

â&#x2C6;&#x2019; 3 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;3

Find x-intercepts:

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;1

x 1

0 1r

x 1

1 8 2 2

Not a real number No x-intercepts

124

NOT FOR SALE

Chapter 2

33. f x

Polynomial ynomial and Rational Function Functions

37. g x

2 x 12

1 x2 4 1 4

1 4

8 x 16

1 4

4 16

x 8 x 48

x 4 x 12

4 or x

34. f x

â&#x2C6;&#x2019;12 â&#x2C6;&#x2019;16

13 x 92

Vertex: 92 , 34

81 4

39. f x

â&#x2C6;&#x2019; 15

â&#x2C6;&#x2019;2

x-intercepts: 4, 0

2 6

40. f x

4 x 2 24 x 41 4 x 2 6 x 41 4 x 2 6 x 9 36 41

4 x 3 5 2

Axis of symmetry: x

3 â&#x2C6;&#x2019; 20

No x-intercepts

â&#x2C6;&#x2019;8

x-intercepts: 3, 0 , 1, 0

41. g x

1 2

4 x 2

Axis of symmetry: x

x 2 x 30

x-intercepts: 2 r

x x 2

Axis of symmetry: x

1 4

42. f x

121 4 35

â&#x2C6;&#x2019; 10

x-intercepts: 6, 0 , 5, 0

x2 3 2 x 5

2 3 2

3 5

2 6, 0

â&#x2C6;&#x2019;8

â&#x2C6;&#x2019;4

6 x 5

3 5

6 x 9 3 2

Vertex: 3, 42 5 â&#x2C6;&#x2019; 80

x 2 x 30

1 4

1 2

Vertex: 2, 3

â&#x2C6;&#x2019;5

Vertex: 3, 5

x 1 4 5

Axis of symmetry: x

10 â&#x2C6;&#x2019;1

x 4 â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;6

x 2 2 x 3

x 12

Vertex: 12 , 121 4

Axis of symmetry: x

9 2

Vertex: 1, 4

36. f x

11, 0

Vertex: 4, 0

x-intercepts: 3, 0 , 6, 0

35. f x

2 x 2 16 x 32

2 x 4

â&#x2C6;&#x2019;4

3 x 6

2 x 2 8 x 16

13 814 6

x 2 9 x 18

5 â&#x2C6;&#x2019; 20

â&#x2C6;&#x2019;2

3x 6

x-intercepts: 5 r

Find x-intercepts: 13 x 2

5 11

Axis of symmetry: x

3 4

Axis of symmetry: x

10 x 25 25 14

3x 6

13 x 2 9 x

â&#x2C6;&#x2019; 18

Vertex: 5, 11

â&#x2C6;&#x2019;20

13 x 2 9 x 6

x 2 10 x 14

x-intercepts: 4, 0 , 12, 0

13 x 2

38. f x

5, 0

â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;8

2 x 12

Find x-intercepts: 1 2 x 4

x-intercepts: 4 r y

Axis of symmetry: x

Vertex: 4, 16

4 5

Vertex: 4, 5

Axis of symmetry: x

x 2 8 x 11

42 5

Axis of symmetry: x

27 5

â&#x2C6;&#x2019; 14

INSTRUCTOR USE ONLY xx-intercepts: intercepts: 3 r

114, 0

â&#x2C6;&#x2019; 10 1

NOT FOR SALE Section 2.1

43. 1, 4 is the vertex.

f x

Because the graph passes through 1, 0 , a 1 1 4 2

a x 4 1 2

Because the graph passes through 2, 3 , 3

a 2 4 1 2

4a 1

1 x 1 4 2

So, y

x 1 4. 2

So, f x

44. 2, 1 is the vertex.

f x

4 1. 2

49. 1, 2 is the vertex.

a x 2 1 2

Because the graph passes through 0, 3 ,

f x

a x 1 2 2

Because the graph passes through 1, 14 ,

a 0 2 1

4a 1

a 1 1 2

4a 2

So, y

2 1. 2

Because the graph passes through 1, 0 , 2

2 x 2 2. 2

46. 2, 0 is the vertex.

f x

a x 2 0 2

a x 2

a 3 2

a 0 2 3 2

4a 3

a. 14 x 2 3. 2

51. 5, 12 is the vertex.

f x

a x 5 12 2

Because the graph passes through 7, 15 ,

2 x 2 . 2

47. 2, 5 is the vertex.

f x

a x 2 3

So, f x

Because the graph passes through 3, 2 ,

So, y

f x

So, y

Because the graph passes through 0, 2 ,

a 1 2 2

4 x 1 2.

50. 2, 3 is the vertex.

a x 2 2

So, f x

45. 2, 2 is the vertex.

125

48. 4, 1 is the vertex.

a x 1 4

Quadratic Functions Function and Models

a x 2 5

a 7 5 12 2

4a Â&#x; a

Because the graph passes through 0, 9 , 9

a 0 2 5

So, f x

3 4

3 . 4

5 12. 2

So, f x

1 x 2 5 2

2 5. 2

126

NOT FOR SALE

Chapter 2

Polynomial ynomial and Rational Function Functions

52. 2, 2 is the vertex.

f x

56. 6, 6 is the vertex.

f x

a x 2 2 2

Because the graph passes through 1, 0 , 0

a 1 2 2

a 2

53. 14 ,

3 2

2 x 2 2. 2

a x

1 4

a 2 49 a 16

3 2

24 x 49

1 4

23 .

Because the graph passes through 2, 4 ,

81 a 4 81 a 4

19 4 19 81

3 4

x2 4 x 5

5 or

5 x 1

2x2 5x 3

12 , 0 , 3, 0

2x2 5x 3

2 x

59. f x

x 52

43 .

a x

5 2

1 x 3

0 Â&#x; x

1 2

0 Â&#x; x

x2 4 x

16 3

a 72

16 3

5 2

â&#x2C6;&#x2019;4

x2 4 x

x x 4

â&#x2C6;&#x2019;4

The x-intercepts and the solutions of f x

0 are the

same.

Because the graph passes through

So, f x

450 x 6 6.

x-intercepts: 0, 0 , 4, 0

19 81

x2 4 x 5

x 3

3 4

55. 52 , 0 is the vertex.

f x

2x 1

So, f x

5 2

1 a 100

x-intercepts:

a 2

58. y

52 , 34 is the vertex. f x

a x 52 34 4

1 a 100

1061, 32 ,

x-intercepts: 5, 0 , 1, 0

. 24 49

3 2

57. y

3 2

Â&#x; a

So, f x

54.

1 4

So, f x

Because the graph passes through 2, 0 , 0

1061 6

3 2

450

is the vertex.

f x

Because the graph passes through

So, f x

a x 6 6

16 x 3

72 ,

16 3

60. f x

2 x 2 10 x

x-intercepts: 0, 0 , 5, 0

5 2

. 2

2 x 2 10 x

2 x x 5

0 Â&#x; x 0 Â&#x; x

â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;6

The x-intercepts and the solutions of f x

0 are the

same.

NOT FOR SALE Section 2.1

61. f x

x 2 9 x 18

65. f x

x 2 9 x 18

3 or x

3 x 6

16 â&#x2C6;&#x2019;4

g x

x 2 2 x 3

0 are the

Note: f x

x-intercepts: 2, 0 , 10, 0

x 2 8 x 20

66. f x

0 Â&#x; x

x 10

0 Â&#x; x

x 25, opens upward

10 0 are the

same.

g x

f x , opens downward

g x

x 2 25

2 x 7 x 30

2 x

and 5, 0 for all real numbers a z 0. 10 â&#x2C6;&#x2019;5

67. f x

g x

â&#x2C6;&#x2019; 40

x 0 x 10

opens downward

x 10 x

Note: f x

0 are the

a x 0 x 10

ax x 10 has

x-intercepts 0, 0 and 10, 0 for all real 7 10

numbers a z 0.

12 x 45

x-intercepts: 15, 0 , 3, 0

12 x 45

15 x 3

0 Â&#x; x 0 Â&#x; x

68. f x

10 â&#x2C6;&#x2019; 18

â&#x2C6;&#x2019; 60

4 x 8

x 12 x 32, opens upward 2

g x

f x , opens downward

g x

x 2 12 x 32

Note: f x

The x-intercepts and the solutions of f x

same.

opens upward

same.

x 3

0 x 10

x 10 x

The x-intercepts and the solutions of f x

x 15

x 2

5 x 6

or x

a x 2 25 has x-intercepts 5, 0

Note: f x

x-intercepts: 52 , 0 , 6, 0

5 x 5

â&#x2C6;&#x2019; 40

The x-intercepts and the solutions of f x

63. f x

ÂŞÂŹ x 5 ÂşÂź x 5

2 x 10

x 2

a x 1 x 3 has x-intercepts 1, 0

and 3, 0 for all real numbers a z 0.

10 â&#x2C6;&#x2019;4

ÂŞÂŹ x 1 ÂşÂź x 3 opens downward x 1 x 3

x 2 8 x 20

1 x 3

x2 2x 3

62. f x

7 10

opens upward

x 2x 3 â&#x2C6;&#x2019;8

same.

127

The x-intercepts and the solutions of f x

64. f x

ÂŞÂŹ x 1 ÂşÂź x 3

x-intercepts: 3, 0 , 6, 0

Quadratic Functions Function and Models

a x 4 x 8 has x-intercepts 4, 0

and 8,0 for all real numbers a z 0.

0 are the 69. f x

ÂŞÂŹ x 3 ÂşÂź ÂŞ x 12 Âş 2

ÂŹ Âź

3 x

1 2

opens upward

3 2 x 1

2x 7x 3 2

g x

2 x 2 7 x 3

opens downward

= 2x2 7 x 3 Note: f x

a x 3 2 x 1 has x-intercepts

INSTRUCTOR USE E ONLY 3,, 0

and 12 , 0 for all real numbers a z 0.

128

Chapter 2

Polynomial ynomial and Rational Function Functions

2 x 52 x 2

2 x 12 x 5

70. f x

2 ÂŞ x 52 Âş x 2

ÂŹ Âź

73. Let x

x 2y

g x

f x , opens downward

g x

2 x 2 x 10

5 2

P x

x 2 has x-intercepts

52 , 0 and 2, 0 for all real numbers a z 0. 71. Let x

the first number and y

the second number.

Then the sum is

x y

110 Â&#x; y

The product is P x

P x

110 x. xy

1 x2 24 x

2 1 x 2 24 x 144 144

2 1ÂŞ 1 2 2 x 12 144Âş x 12 72 Âź 2ÂŹ 2

24 12

6. So, the numbers are 12 and 6.

110 x x 2 .

74. Let x

x 2 110 x

the first number and y

x 110 x 3025 3025

Then the sum is x 3 y

2 ÂŞ x 55 3025Âş ÂŹ Âź

The product is P x

x 55 3025

The maximum value of the product occurs at the vertex of P x and is 3025. This happens when x y 55. 72. Let x the first number and y Then the sum is

x y

S Â&#x; y

P x

the second number.

S x.

The product is P x

x S x

Â§ 24 x Âˇ xÂ¨ Â¸. ÂŠ 2 Âš

The maximum value of the product occurs at the vertex of P x and is 72. This happens when x 12 and y

x 110 x

24 x . 2

24 Â&#x; y

The product is P x

2 x 2 x 10, opens upward

a x

the second number.

Then the sum is

Note: f x

the first number and y

Sx x 2 .

42 Â&#x; y

42 x . 3

Â§ 42 x Âˇ xÂ¨ Â¸. ÂŠ 3 Âš

1 x2 42 x

3 1 x 2 42 x 441 441

3 1ÂŞ 1 2 2 x 21 441Âş x 21 147 Âź 3ÂŹ 3

The maximum value of the product occurs at the vertex of P x and is 147. This happens when x 21 and y

Sx x 2

the second number.

42 21 3

7. So, the numbers are 21 and 7.

x 2 Sx Â§ S2 S2 Âˇ Â¨ x 2 Sx Â¸ 4 4 Âš ÂŠ

75. y

The vertex occurs at

SÂˇ S2 Â§ Â¨ x Â¸ 2Âš 4 ÂŠ

The maximum value of the product occurs at the vertex of P x and is S 2 4. This happens when x

4 24 x2 x 12 9 9

b 2a

24 9 2 4 9

maximum height is 4 2 24 y 3

3 3 12 9 9

3. The

16 feet.

S 2.

NOT FOR SALE Section 2.1

76. y

Quadratic Functions Function and Models

129

16 2 9 x x 1.5 2025 5

(a) The ball height when it is punted is the y-intercept.

16 9 2 0 0 1.5 2025 5

(b) The vertex occurs at x

1.5 feet

b 2a

Â§ 3645 Âˇ The maximum height is f Â¨ Â¸ ÂŠ 32 Âš

95 2 16 2025

3645 . 32 2

16 Â§ 3645 Âˇ 9 Â§ 3645 Âˇ Â¨ Â¸ Â¨ Â¸ 1.5 2025 ÂŠ 32 Âš 5 ÂŠ 32 Âš

6561 6561 1.5 64 32

6561 13,122 96 64 64 64

6657 feet | 104.02 feet. 64

16 2 9 x x 1.5 2025 5

9 5 r

9 5 2

4 1.5 16 2025

32 2025

1.8 r 1.81312 0.01580247

x | 0.83031 or x | 228.64

The punt is about 228.64 feet. 800 10 x 0.25 x 2

77. C

0.25 x 2 10 x 800

The vertex occurs at x

b 2a

The cost is minimum when x

10 2 0.25

20.

20 fixtures.

230 20 x 0.5 x 2

78. P

20 2 0.5

20.

Because x is in hundreds of dollars, 20 u 100 2000 dollars is the amount spent on advertising that gives maximum profit. 79. R p

12 p 2 150 p

(a) R $4

12 $4 150 $4

$408

R $6

12 $6 150 $6

$468

R $8

12 $8 150 $8

$432

2 2

(b) The vertex occurs at b 2a

The vertex occurs at x

80. R p

b 2a

(a) R 20

$14,000 thousand

$14,000,000

R 25

$14,375 thousand

$14,375,000

R 30

$13,500 thousand

$13,500,000

150 2 12

$6.25.

Revenue is maximum when price

$6.25 per pet.

The maximum revenue is R $6.25

25 p 2 1200 p

12 $6.25 150 $6.25

$468.75.

(b) The revenue is a maximum at the vertex.

b 2a

R 24

1200 2 25

14,400

The unit price that will yield a maximum revenue of $14,400 thousand is $24.

130

NOT FOR SALE

Chapter 2

Polynomial ynomial and Rational Function Functions

81. (a) y

1 200 4 x

4x 3y

200 Â&#x; y

Âª4 Âº 2 x Â« 50 x Â» Â¬3 Â¼

(b)

2 xy

x 5 10

(d) A

600 1066

2 3

1400

1600

8 x 50 x

(c)

4 50 x

3 8 x 50 x

1666

This area is maximum when x 100 1 and y 33 feet. 3 3

2000

25 feet

This area is maximum when x 100 1 and y 33 feet. 3 3

2 3

1600 8 x 50 x

3 8 x 2 50 x

3 8 2 x 50 x 625 625

3 8Âª 2 x 25 625Âº Â¼ 3Â¬ 8 5000 2 x 25 3 3

The maximum area occurs at the vertex and is 5000 3 square feet. This happens when x y

25 feet

200 4 25

100 3 feet. The dimensions are 2 x

25 feet and

1 50 feet by 33 feet. 3

(e) They are all identical. x

25 feet and y

1 33 feet 3

NOT FOR SALE Section 2.1

82. (a)

Quadratic Functions Function and Models

131

(b) Radius of semicircular ends of track: r

1 y 2

2S r

Distance around two semicircular parts of track: d (c) Distance traveled around track in one lap: d

S y 2x Sy y

Â§1 Âˇ 2S Â¨ y Â¸ ÂŠ2 Âš

200 200 2 x 200 2 x

(d) Area of rectangular region: A

Â§ 200 2 x Âˇ xÂ¨ Â¸ S ÂŠ Âš 1 200 x 2 x 2

S 2

100 x

100 x 2500 2500

50 2

5000

The area is maximum when x

50 and y

200 2 50

100

number of tickets sold price per ticket

83. (a) Revenue

Let y

attendance, or the number of tickets sold. 100, 20, 1500

y 1500

100 x 20

y 1500

100 x 2000

100 x 3500

R x

y x

100 x

R x

100 x 3500 x

3500 x

(b) The revenue is at a maximum at the vertex.

b 2a

R 17.5

3500 2 100

17.5

100 17.5 3500 17.5

$30,625

A ticket price of $17.50 will yield a maximum revenue of $30,625.

132

NOT FOR SALE

Chapter 2

Polynomial ynomial and Rational Function Functions

84. (a) Area of window

Area of rectangle Area of semicircle

1 2 S radius

1 Â§ xÂˇ SÂ¨ Â¸ 2 ÂŠ2Âš

Sx2 8

To eliminate the y in the equation for area, introduce a secondary equation. perimeter of rectangle perimeter of semicircle

Perimeter

1 circumference

2 1 2 y x 2S Â&#x2DC; radius

2 Â§ xÂˇ 2y x S Â¨ Â¸ ÂŠ 2Âš

2y x

16 16 16

1 Sx x 2 4

Substitute the secondary equation into the area equation. xy

Area

S x2

1 S x Âˇ S x2 Â§ xÂ¨ 8 x Â¸ 2 4 Âš 8 ÂŠ

1 2 S x2 S x2 x 2 4 8

1 2 S x2 x 2 8

1 64 x 4 x2 S x2

(b) The area is maximum at the vertex. 8x

Area

Â§ 1 SÂˇ 2 Â¨ Â¸ x 8x 8Âš ÂŠ 2

8 | 4.48 Â§ 1 SÂˇ 2Â¨ Â¸ 8Âš ÂŠ 2

b 2a

1 2 S x2 x 2 8

S 4.48

1 | 2.24 4.48 2 4

The area will be at a maximum when the width is about 4.48 feet and the length is about 2.24 feet. 85. (a)

4200

55 0

(b) The maximum annual consumption occurs at the point 16.9, 4074.813 . 4075 cigarettes 1966 o t

The maximum consumption occurred in 1966. After that year, the consumption decreases. It is likely that the warning was responsible for the decrease in consumption. (c) Annual consumption per smoker

Annual consumption in 2005 Â&#x2DC; total population total number of smokers in 2005

1487.9 296,329,000

59,858,458

7365.8

About 7366 cigarettes per smoker annually Daily consumption per smoker

Number of cigarettes per year Number of days per year

7366 | 20.2 365

About 20 cigarettes per day

NOT FOR SALE Section 2.1

86. (a) and (c)

90. f x

The model fits the data well. 0.0666 x 2 0.875 x 2.69

(d) 2006 (e) Answers will vary. (f ) Let t

0.0666 13 0.875 13 2.69 | 2.81

In the year 2013, the sales will be about $2.81 billion. 87. True. The equation 12 x 2 1 0 has no real solution, so the graph has no x-intercepts.

88. True. The vertex of f x is 54 ,

g x is 89. f x

54 ,

71 4

53 4

and the vertex of

x 2 bx 75, maximum value: 25

The maximum value, 25, is the y-coordinate of the vertex. Find the x-coordinate of the vertex: x

b 2a f x

Â§bÂˇ fÂ¨ Â¸ ÂŠ 2Âš 25

b 2 1

b 2

Â§bÂˇ Â§bÂˇ Â¨ Â¸ bÂ¨ Â¸ 75 ÂŠ 2Âš ÂŠ 2Âš b2 b2 75 4 2

400

b2 4 b2

r20

100

f x

Â§bÂˇ fÂ¨ Â¸ ÂŠ 2Âš 48

x 2 bx 16 2

Â§bÂˇ Â§bÂˇ Â¨ Â¸ bÂ¨ Â¸ 16 ÂŠ 2Âš ÂŠ 2Âš

b2 b2 16 4 2

256

b2 4 b2

r16

91. f x

x 2 bx 26 , minimum value: 10

The minimum value, 10, is the y-coordinate of the vertex. Find the x-coordinate of the vertex: b b b x 2a 2 1

2 f x

Â§ bÂˇ f Â¨ Â¸ ÂŠ 2Âš

x 2 bx 26 2

Â§ bÂˇ Â§ bÂˇ Â¨ Â¸ bÂ¨ Â¸ 26 2 ÂŠ Âš ÂŠ 2Âš

b2 b2 26 4 2 b2 4 b2

x 2 bx 75

x 2 bx 16, maximum value: 48

13, y

133

The maximum value, 48, is the y-coordinate of the vertex. Find the x-coordinate of the vertex: b b b x 2a 2 1

(b) y

Quadratic Functions Function and Models

92. f x

x 2 bx 25 , minimum value: â&#x20AC;&#x201C;50

The minimum value, â&#x20AC;&#x201C;50, is the y-coordinate of the vertex. Find the x-coordinate: b b b x 2a 2 1

2 f x

Â§ bÂˇ f Â¨ Â¸ ÂŠ 2Âš

x 2 bx 25 2

Â§ bÂˇ Â§ bÂˇ Â¨ Â¸ bÂ¨ Â¸ 25 ÂŠ 2Âš ÂŠ 2Âš

100

b2 b2 25 4 2 b 2 4 b2

r10

50 25

134

Chapter 2

93. f x

NOT FOR SALE

Polynomial ynomial and Rational Function Functions

ax 2 bx c

94. (a) Since the graph of P opens upward, the value of a is positive.

b Âˇ Â§ aÂ¨ x 2 x Â¸ c a Âš ÂŠ

(b) Since the graph of P opens upward, the vertex of the b parabola is a relative minimum at t . 2a

Â§ b b2 b2 Âˇ aÂ¨ x 2 x Â¸ c 2 a 4a 4a 2 Âš ÂŠ 2

(c) Because of the symmetrical property of the graph of a parabola, and since the company made the same yearly profit in 2004 and 2012, the midpoint of the interval 4 d t d 12 or t 8 corresponds to the year 2008,when the company made the least profit.

b Âˇ b Â§ aÂ¨ x c Â¸ 2a Âš 4a ÂŠ 2

b Âˇ 4ac b 2 Â§ aÂ¨ x Â¸ 2a Âš 4a ÂŠ

(d) Since the year 2008 is when the company made the least profit, profit has been increasing since 2008, and is also currently increasing.

Â§ b Âˇ Â§ b Âˇ a Â¨ 2 Â¸ bÂ¨ Â¸ c a 4 ÂŠ 2a Âš ÂŠ Âš

Â§ b Âˇ f Â¨ Â¸ ÂŠ 2a Âš

b2 b2 c 4a 2a b 2 2b 2 4ac 4a So, the vertex occurs at Â§ b 4ac b 2 Âˇ Â§ b Â¨ , Â¸ Â¨ , 2 a 4 a ÂŠ 2a ÂŠ Âš 95. If f x

Â§ b ÂˇÂˇ f Â¨ Â¸ Â¸. ÂŠ 2a Âš Âš

ax 2 bx c has two real zeros, then by the Quadratic Formula they are

b r

4ac b 2 4a

b 2 4ac . 2a

The average of the zeros of f is b

b 2 4ac b 2a 2

b 2 4ac 2a

2b 2a 2

b . 2a

This is the x-coordinate of the vertex of the graph.

Section 2.2 Polynomial Functions of Higher Degree 10. f x

1. continuous

2 x3 3 x 1 has intercepts

2. Leading Coefficient Test

0, 1 , 1, 0 , 12

3. n; n 1

Matches graph (f).

4. (a) solution; (b) x a ; (c) x-intercept 5. touches; crosses 6. multiplicity

11. f x

r2 12. f x

7. standard

1 2

3, 0 and 12

1 2

3, 0 .

14 x 4 3 x 2 has intercepts 0, 0 and

3, 0 . Matches graph (a).

13 x3 x 2

4 3

has y-intercept 0, 43 .

Matches graph (e).

8. Intermediate Value

13. f x

9. f x

2 x 2 5 x is a parabola with x-intercepts

0, 0

and 52 , 0 and opens downward. Matches

graph (h).

x 4 2 x3 has intercepts 0, 0 and 2, 0 .

Matches graph (d). 14. f x

1 x5 5

2 x3

9 x 5

has intercepts

0, 0 , 1, 0 , 1, 0 , 3, 0 , 3, 0 .

Matches graph (b).

NOT FOR SALE Section 2.2

15. y

Polynomial Functions of Hi Higher Degree H x5

16. y

(a) f x

135

(a) f x

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;4

Horizontal shift four units to the right (b) f x

Horizontal shift one unit to the left (b) f x

x3 4

x5 1

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019; 3 â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;6

Vertical shift four units downward (c) f x

Vertical shift one unit upward (c) f x

1 x3 4

1 x5 2

y y

2 1 2

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019; 3 â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;4

Reflection in the x-axis and a vertical shrink each y -value is multiplied by 14

(d) f x

4 4 3

Reflection in the x-axis, vertical shrink each y -value is multiplied by 12 , and vertical shift one unit upward

(d) f x

12 x 1

y 2

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;5 â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;5

â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;4

Horizontal shift four units to the right and vertical shift four units downward

Reflection in the x-axis, vertical shrink each y -value is multiplied by 12 , and horizontal shift one unit to the left

136

NOT FOR SALE

Chapter 2

Polynomial ynomial and Rational Function Functions

17. y

(a) f x

(b)

f x

x4 3

4 x4 y

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019; 3 â&#x2C6;&#x2019;2

â&#x2C6;&#x2019; 5 â&#x2C6;&#x2019; 4 â&#x2C6;&#x2019; 3 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;1

(e)

f x

2 x 4

â&#x2C6;&#x2019;2

Vertical shift three units downward

â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;4

Horizontal shift three units to the left

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;2

1 2

f x

(d) f x

(c)

Reflection in the x-axis and then a vertical shift four units upward

(f )

12 x

f x

5 4 3 2 1

â&#x2C6;&#x2019; 4 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019; 2 â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;2

(a)

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019; 3

â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;2

Horizontal shift one unit to the right and a vertical shrink each y -value is multiplied by

18. y

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019; 3 â&#x2C6;&#x2019; 2 â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019;1

1 2

Vertical shift two units downward and a horizontal stretch each y -value

Vertical shift one unit upward and a horizontal shrink each y -value is multiplied by 16

1 is multipied by 16

x6 f x

18 x 6

(b)

f x

2 4 6

(c)

x6 5 y

1 â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2

f x

x 2

â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;5 â&#x2C6;&#x2019; 4

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;2

x â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;4

Vertical shrink each y -value is multiplied by

1 8

Horizontal shift two units to the left and a vertical shift four units downward

and

reflection in the x-axis

Vertical shift five units downward

NOT FOR SALE Section 2.2

(d) f x

14 x 6 1

(e)

f x

14 x

Polynomial Functions of Hi Higher Degree H

(f )

f x

2 x 6

137

4 3 2

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2

x 2

â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019; 8 â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;4

x 2

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019; 3 â&#x2C6;&#x2019; 2 â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;4

Reflection in the x-axis, vertical shrink each y -value is multiplied by

â&#x2C6;&#x2019;2

1 4

, and

x 1

â&#x2C6;&#x2019;2

Horizontal stretch each x-value is multiplied by 4 , and vertical shift two units downward

Horizontal shrink each x -value

is multiplied by 12 , and vertical shift one unit downward

vertical shift one unit upward

19. f x

1 x3 5

24. g x

Degree: 4

Degree: 3 Leading coefficient:

Leading coefficient: 1

1 5

The degree is odd and the leading coefficient is positive. The graph falls to the left and rises to the right. 20. f x

x4 4 x 6

The degree is even and the leading coefficient is negative. The graph falls to the left and falls to the right. 25. f x

2 x 2 3x 1

2.1x5 4 x3 2

Degree: 5

Degree: 2

Leading coefficient: 2.1

Leading coefficient: 2 The degree is even and the leading coefficient is positive.

The degree is odd and the leading coefficient is negative.

The graph rises to the left and rises to the right.

The graph rises to the left and falls to the right.

21. g x

7 x 2

26. f x

3x 2

Degree: 5

Degree: 2

Leading coefficient: 4

Leading coefficient: 3 The degree is even and the leading coefficient is negative. The graph falls to the left and falls to the right. 22. h x

The degree is odd and the leading coefficient is positive. The graph falls to the left and rises to the right. 27. f x

1 x6

6 2 x 4 x 2 5 x3

Degree: 3

Degree: 6

Leading coefficient: 5

Leading coefficient: 1 The degree is even and the leading coefficient is negative. The graph falls to the left and falls to the right. 23. g x

4 x5 7 x 6.5

x3 3x 2

The degree is odd and the leading coefficient is negative. The graph rises to the left and falls to the right. 28. f x

Degree: 3

3x 4 2 x 5 4

Leading coefficient: 1

Degree: 4

The degree is odd and the leading coefficient is negative. The graph rises to the left and falls to the right.

Leading coefficient:

3 4

The degree is even and the leading coefficient is positive. The graph rises to the left and rises to the right.

138

NOT FOR SALE

Chapter 2

29. h x

Polynomial ynomial and Rational Function Functions

35. f x

3 2 t 3t 6

(a) 0

Degree: 2 Leading coefficient:

3 4

x 2 36

6 x 6

x 6

78 s 3 5s 2 7 s 1

x 6

The degree is even and the leading coefficient is negative. The graph falls to the left and falls to the right. 30. f s

x 2 36

Zeros: r6 (b) Each zero has a multiplicity of one (odd multiplicity).

Degree: 3

Leading coefficient:

(d)

6 â&#x2C6;&#x2019; 12

The degree is odd and the leading coefficient is negative. The graph rises to the left and falls to the right. 31. f x

3x3 9 x 1; g x

â&#x2C6;&#x2019; 42

3x3

â&#x2C6;&#x2019;4

36. f x

81 x 2

(a) 0

81 x 2

0 â&#x2C6;&#x2019;8

13 x3 3 x 2 , g x

32. f x

13 x3

x 9 x

9 x

0 9

Zeros: r 9

(b) Each zero has a multiplicity of one (odd multiplicity).

g f â&#x2C6;&#x2019;9

â&#x2C6;&#x2019;6

33. f x

x 4 4 x3 16 x ; g x

x4 â&#x2C6;&#x2019; 15

15 â&#x2C6;&#x2019;9

â&#x2C6;&#x2019;8

37. h t

g f

(a) 0

â&#x2C6;&#x2019; 20

34. f x

3x 4 6 x 2 , g x

3x4

t 2 6t 9

Zero: t (b) t

3 has a multiplicity of 2 (even multiplicity).

â&#x2C6;&#x2019;6

t 2 6t 9

(d)

â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;2

NOT FOR SALE Section 2.2

38. f x

x 2 10 x 25

(a) 0

x 2 10 x 25

41. f x

(a) 0

Zero: x (b) x

Polynomial Functions of Hi Higher Degree H 3 x3 12 x 2 3 x

3 x x 2 4 x 1

3 x3 12 x 2 3x

Zeros: x

5 has a multiplicity of 2 (even multiplicity).

139

0, x

3 (by the Quadratic

Formula)

(b) Each zero has a multiplicity of 1 (odd multiplicity).

(d)

(d) â&#x2C6;&#x2019;6 â&#x2C6;&#x2019; 25

â&#x2C6;&#x2019;5

39. f x

(a) 0

1 x2 3 1 2 x 3

13 x

1 3

13 x

â&#x2C6;&#x2019; 24

2 3

x 2

(a) 0 0

2 x 1

2, x

Zeros: x

5 x x 2 2 x 1

42. g x

2 3

0, a

1 ,b 2

5 ,c 2

3 . 2

(d) â&#x2C6;&#x2019;6

Â§5Âˇ Â§ 1 ÂˇÂ§ 3 Âˇ Â¨ Â¸ 4Â¨ Â¸Â¨ Â¸ ÂŠ 2Âš ÂŠ 2 ÂšÂŠ 2 Âš 1

5 r 2

37 4

â&#x2C6;&#x2019; 16

43. f t

t 3 8t 2 16t

(a) 0

5 r 37 2

(b) Each zero has a multiplicity of 1 (odd multiplicity). (c) Turning points: 1 (the vertex of the parabola) (d)

0, x

5 r 2

Zeros: x

4 1 1

1 2 5 3 x x 2 2 2

(b) Each zero has a multiplicity of 1 (odd multiplicity).

1 2 5 3 x x 2 2 2

2 1

2, c

1, b

Zeros: x

â&#x2C6;&#x2019;4

(a) For

2 r

0, a

2r 8 2

â&#x2C6;&#x2019;6

40. f x

x x 2 2 x 1

For x 2 2 x 1

(b) Each zero has a multiplicity of 1 (odd multiplicity). (d)

5 x x 2 2 x 1

t 3 8t 2 16t

t t 2 8t 16

t t 4 t 4

t 4

Zeros: t â&#x2C6;&#x2019;8

t 4

0, t

0 is 1 (odd multiplicity).

(b) The multiplicity of t

4 is 2 (even multiplicity).

The multiplicity of t â&#x2C6;&#x2019;5

(d)

INSTRUCTOR USE ONLY â&#x2C6;&#x2019;9

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019; 2

140

NOT FOR SALE

Chapter 2

44. f x

Polynomial ynomial and Rational Function Functions

47. f x

x 4 x3 30 x 2

(a) 0

3x 4 9 x 2 6 3x 4 9 x 2 6

x 2 x 2 x 30

3 x 4 3 x 2 2

x 2 x 6 x 5

3 x 2 1 x 2 2

x 6

x 5

Zeros: x

0, x

(b) No real zeros

6, x

0 is 2 (even multiplicity).

(b) The multiplicity of x The multiplicity of x

6 is 1 (odd multiplicity).

The multiplicity of x

5 is 1 (odd multiplicity).

60 â&#x2C6;&#x2019;9

â&#x2C6;&#x2019;6

48. f x

2 x 4 2 x 2 40

(a) 0

2 x 4 2 x 2 40

2 x 4 x 2 20

2 x 2 4 x 2 5

Zeros: t (b) t t

t t 4 6t 2 9

t 5 6t 3 9t tt

t 2

0, t

Zeros: x

t 5 6t 3 9t

(a) 0

6 â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;300

45. g t

t t 2 3

(b) Each zero has a multiplicity of 1 (odd multiplicity). (c) Turning points: 3 (d)

20 â&#x2C6;&#x2019;6

0 has a multiplicity of 1 (odd multiplicity). r

49. g x

multiplicity). (c) Turning points: 4 (d)

â&#x2C6;&#x2019; 60

3 each have a multiplicity of 2 (even

(a) 0

x3 3 x 2 4 x 12

â&#x2C6;&#x2019;9

4 x 3

r 2, x

Zeros: x

x 2 x 3 4 x 3

2 x 2 x 3

(b) Each zero has a multiplicity of 1 (odd multiplicity). â&#x2C6;&#x2019;6

46. (a) f x

x5 x3 6 x

x x 4 x 2 6

x x 2 3 x 2 2

0, r

Zeros: x

(d)

4 â&#x2C6;&#x2019;8

â&#x2C6;&#x2019;16

(b) Each zero has a multiplicity of 1 (odd multiplicity). (c) Turning points: 2 (d)

â&#x2C6;&#x2019;9

â&#x2C6;&#x2019;6

NOT FOR SALE Section 2.2

50. f x

x3 4 x 2 25 x 100

(a) 0

x 2 x 4 25 x 4

x 5 x 5 x 4

Polynomial Functions of Higher Hi H Degree x5 5 x3 4 x

53. y

(a)

25 x 4

â&#x2C6;&#x2019;6

r 5, 4

Zeros: x

â&#x2C6;&#x2019;4

(b) Each zero has a multiplicity of 1 (odd multiplicity).

(b) x-intercepts: 0, 0 , r1, 0 , r2, 0

(d)

â&#x2C6;&#x2019;9

â&#x2C6;&#x2019; 20

51. y

x5 5 x3 4 x

x x 2 1 x 2 4

x x 1 x 1 x 2 x 2

0, r1, r 2

140

(d) The solutions are the same as the x-coordinates of the x-intercepts.

4 x 20 x 25 x 3

(a)

1 x3 4

54. y

(a) â&#x2C6;&#x2019;2

141

6 â&#x2C6;&#x2019; 18

â&#x2C6;&#x2019;4

(b) x-intercepts: 0, 0 ,

52 , 0

â&#x2C6;&#x2019;12

4 x3 20 x 2 25 x

(b) x-intercepts: 0, 0 , 3, 0 , 3, 0

x 4 x 2 20 x 25

x 2 x 5

1 x3 4

0, r 3

(d) The solutions are the same as the x-coordinates of the x-intercepts.

5 2

(d) The solutions are the same as the x-coordinates of the x-intercepts.

55. f x

0 x 8

x 8x 2

52. y

4 x3 4 x 2 8 x 8

Note: f x

(a) â&#x2C6;&#x2019;3

ax x 8 has zeros 0 and 8 for all real

numbers a z 0.

56. f x

0 x 7

x 7x 2

â&#x2C6;&#x2019; 11

(b)

1, 0 ,

2, 0

Note: f x

57. f x

4 x x 1 8 x 1

0 x

ax x 7 has zeros 0 and 7 for all real

numbers a z 0.

4 x3 4 x 2 8 x 8

4 x 4 x 2

2, 0 ,

2 x 6

x 2 4 x 12

8 x 1

Note: f x

2 x 1

all real numbers a z 0. 58. f x

2, 1

(d) The solutions are the same as the x-coordinates of the x-intercepts.

a x 2 x 6 has zeros 2 and 6 for

4 x 5

x x 20 2

Note: f x

a x 4 x 5 has zeros 4 and 5 for

INSTRUCTOR USE ONLY all real numbers a z 0.

142

Chapter 2

59. f x

NOT FOR SALE

Polynomial ynomial and Rational Function Functions

0 x 4 x 5

64. f x

x x 9 x 20

x3 9 x 2 20 x Note: f x

ax x 1 x 10 has zeros 0, 1, and

10 for all real numbers a z 0.

65. f x

a x 4 4 x3 9 x 2 36 x has zeros

x 2

x 1

x 0 x 1 x 2

66. f x

x2 6 x 9

a x 2 6 x 9 , a z 0, has degree 2 and

12 x 6

x 2 18 x 72

Note: f x

a x 2 18 x 72 , a z 0, has degree 2

and zeros x

12 and 6.

67. f x

0 x 5 x 1

x x 2 4 x 5

x x 4 x 1

5 for all real numbers a z 0.

3 x 3

zero x

x x 2 x 1 x 1 x 2

x3 4 x 2 5 x

x x 4 5 x 2 4

Note: f x

ax x 2 4 x 5 , a z 0, has degree 3

x 5 5 x3 4 x

and zeros x

0, 5, and 1.

Note: f x

ax x 2 x 1 x 1 x 2 has

zeros 2, 1, 0, 1, and 2 for all real numbers a z 0.

ÂŞx 1 ÂŹ

3 ÂşÂŞ x 1 ÂźÂŹ

ÂŞ x 1 ÂŹ

3 ÂşÂŞ ÂźÂŹ x 1

1 2

68. f x

Note: f x

3 ÂşÂź

degree 3 and zeros x

69. f x

x 2x 2 a x 2 2 x 2 has zeros

3 and 1

3 x 3

3 x

x3 3x

a x3 3x , a z 0, has degree 3 and

Note: f x

zeros x

0 x

x x

2, 4, and 7.

x3 9 x 2 6 x 56

a x3 9 x 2 6 x 56 , a z 0, has

x 2x 1 3 Note: f x

x 2 x 4 x 7

x 2 x 2 11x 28

3Âş Âź

4 x 2 9 x

4, 3, 3, and 0 for all real numbers a z 0.

63. f x

5, and 4

Note: f x

x 4 4 x3 9 x 2 36 x

62. f x

2, 4

4 x 3 x 3 x 0

Note: f x

a x3 10 x 2 27 x 22 has zeros

Note: f x

x 11x 10 x

x x

2 2 ÂŞ x 4 5Âş ÂŹ Âź

x3 10 x 2 27 x 22

x x 11x 10

61. f x

5Âş Âź 5 ÂşÂź

x3 8 x 2 16 x 5 x 2 x 2 16 x 32 10

Note: f x

5 ÂşÂŞ x 4 ÂźÂŹ 5 ÂşÂŞ ÂźÂŹ x 4

x x 4 5 x 2 x 4 10

0 x 1 x 10

2 ÂŞ x 4 ÂŹ x 2 ÂŞÂŹ x 4

ax x 4 x 5 has zeros 0, 4, and

5 for all real numbers a z 0. 60. f x

3, and

3 for all real numbers

70. f x

x3 27 x 2 243 x 729

Note: f x

a x3 27 x 2 243x 729 , a z 0, has

degree 3 and zero x

x 5

x 1 x 2 x 4 7 x3 3x 2 55 x 50 2 f x x 5

x 1 x 2

x 4 x3 15 x 2 23 x 10 2 f x x 5

x 1 x 2

x 4 17 x 2 36 x 20

71. f x

or or

INSTRUCTOR USE ONLY Note Any nonzero scalar multiple of these functions would also have degree 4 and zeros x Note:

5, 1, and 2.

NOT FOR SALE Section 2.2

72. f x

4 x 1 x 3 x 6

x 4 x 4

4, 1, 3, and 6.

x5 4 x 4

or f x

x 3 x 4

or f x

x 2 x 4

or f x

x x 4

x5 8 x 4 16 x3

x5 12 x 4 48 x3 64 x 2

x5 16 x 4 96 x3 256 x 2 256 x 0 and 4.

Note: Any nonzero scalar multiple of these functions would also have degree 5 and zeros x 74. f x

1 x 4 x 7 x 8

x5 17 x 4 79 x3 11x 2 332 x 224

or f x

1 x 4 x 7 x 8

x5 22 x 4 169 x3 496 x 2 208 x 896

or f x

1 x 4 x 7 x 8

x5 25 x 4 223 x3 787 x 2 532 x 1568

or f x

1 x 4 x 7 x 8

x5 26 x 4 241x3 884 x 2 640 x 1792

1, 4, 7, and 8.

Note: Any nonzero scalar multiple of these functions would also have degree 5 and zeros x 75. f x

x x 5 x 5

x3 25 x

77. f t

1 4

t 2

2t 15

1 4

1 2

(a) Falls to the left; rises to the right

(a) Rises to the left; rises to the right

(b) Zeros: 0, 5, 5

(b) No real zeros (no x-intercepts)

(c)

x f x

(d)

143

x 4 4 x3 23 x 2 54 x 72

a x 4 4 x3 23x 2 54 x 72 , a z 0, has degree 4 and zeros x

Note: f x

73. f x

Polynomial Functions of Higher Hi H Degree

(c)

t f t

7 2

4.5

3.75

3.5

3.75

4.5

72 .

(d) The graph is a parabola with vertex 1,

y 8

(â&#x2C6;&#x2019;5, 0) (5, 0)

(0, 0) â&#x2C6;&#x2019;8 â&#x2C6;&#x2019; 6

â&#x2C6;&#x2019;2

x 4

â&#x2C6;&#x2019;24 â&#x2C6;&#x2019;36

â&#x2C6;&#x2019;48

76. g x

x 2 x 3 x 3

x4 9x2

(a) Rises to the left; rises to the right

x f x

(b) Zeros: 2, 8

(c)

g x

15 10

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;1

(d) (0, 0) 1

(3, 0) 2

x 2 x 8

(a) Falls to the left; falls to the right

x 2 10 x 16

â&#x2C6;&#x2019;4

(d)

(â&#x2C6;&#x2019; 3, 0)

â&#x2C6;&#x2019;2

78. g x

(b) Zeros: 3, 0, 3 (c)

â&#x2C6;&#x2019;4

4 8

â&#x2C6;&#x2019; 15

â&#x2C6;&#x2019;20

â&#x2C6;&#x2019;25

INSTRUCTOR USE ONLY (2, 0) 4

(8, 0)

144

NOT FOR SALE

Chapter 2

79. f x

Polynomial ynomial and Rational Function Functions x 2 x 2

x3 2 x 2

82. f x

4 x3 4 x 2 15 x

(a) Falls to the left; rises to the right

x 4 x 2 4 x 15

(b) Zeros: 0, 2

x 2 x 5 2 x 3

(c)

x f x

1 2

(a) Rises to the left; falls to the right

(b) Zeros: 32 , 0,

(d)

(c)

f x

3 2

(d)

(0, 0) (2, 0)

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019; 3 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019; 1

5 2

16 12 8

â&#x2C6;&#x2019;4

80. f x

(

â&#x2C6;&#x2019; 3, 0 2

8 x3

x 4 2 x x 2

( 52, 0(

(0, 0)

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019; 2

(a) Rises to the left; falls to the right (b) Zero: 2 (c)

83. f x

x f x

(d)

(a) Rises to the left; falls to the right

(b) Zeros: 0, 5 (c)

f x

(d) 6

(â&#x2C6;&#x2019; 5, 0)

â&#x2C6;&#x2019; 15

(2, 0)

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019; 1

81. f x

x 2 5 x

5 x 2 x3

(0, 0)

â&#x2C6;&#x2019; 10

x 4

3x3 15 x 2 18 x

3 x x 2 x 3

â&#x2C6;&#x2019; 20

(a) Falls to the left; rises to the right (b) Zeros: 0, 2, 3 (c)

x f x

(d)

2.5

3.5

1.875

7.875

y 7 6 5 4 3 2

(0, 0) 1 (2, 0) â&#x2C6;&#x2019;3 â&#x2C6;&#x2019; 2 â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019;1

(3, 0) x

â&#x2C6;&#x2019;2

NOT FOR SALE Section 2.2

84. f x

Polynomial Functions of Higher Hi H Degree

145

48 x 2 3 x 4 3x 2 x 2 16

(a) Rises to the left; rises to the right (b) Zeros; 0, r 4 (c)

x f x

(d)

675

189

144

â&#x20AC;&#x201C;144

â&#x20AC;&#x201C;189

675

(â&#x2C6;&#x2019; 4, 0) 100

(0, 0) â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;2

(4, 0)

â&#x2C6;&#x2019;200 â&#x2C6;&#x2019; 300

85. f x

x 2 x 4

87. g t

14 t 2 t 2

(a) Falls to the left; rises to the right

(a) Falls to the left; falls to the right

(b) Zeros: 0, 4

(b) Zeros: 2, 2

(c)

x f x

(c)

g t

25 4

(d)

(0, 0) â&#x2C6;&#x2019;4

(d)

2 â&#x2C6;&#x2019;2

(4, 0)

â&#x2C6;&#x2019;3

(2, 0)

(â&#x2C6;&#x2019; 2, 0) â&#x2C6;&#x2019;1

t 1

â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;5 â&#x2C6;&#x2019;6

86. h x

1 3 x 3

88. g x

(a) Falls to the left; rises to the right

1 x 3

(b) Zeros: 1, 3

h x

25 3

(d)

(a) Falls to the left; rises to the right

(b) Zeros: 0, 4 (c)

1 10

2 32 3

125 3

(c)

g x

12.5

2.7

3.2

0.9

2.5

y 14

(d)

8 6

(â&#x2C6;&#x2019;1, 0)

â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;2

(0, 0) â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;2

(4, 0) 2

(3, 0) 4

x 8

x 8 10 12

146

NOT FOR SALE

Chapter 2

89. f x

Polynomial ynomial and Rational Function Functions x x 4 x 4

x3 16 x

94. f x

0.11x3 2.07 x 2 9.81x 6.88

The function has three zeros. They are in the intervals >0, 1@, >6, 7@, and >11, 12@. They are approximately 0.845,

â&#x2C6;&#x2019;6

6.385, and 11.588.

10 â&#x2C6;&#x2019;32

Zeros: 0 of multiplicity 1; 4 of multiplicity 1; and 4 of

â&#x2C6;&#x2019;4

multiplicity 1 90. f x

â&#x2C6;&#x2019; 10

1 x4 4

â&#x2C6;&#x2019;9

â&#x2C6;&#x2019;6

Zeros: 2.828 and 2.828 of multiplicity 1; 0 of multiplicity 2 91. g x

1 5

1 x 3 2 x 9

â&#x2C6;&#x2019; 12

6.88

1.91

0.97

4.56

5.34

6.07

6.89

5.78

6.28

3.03

4.17

2.84

1.12

95. g x

3x 4 4 x3 3

â&#x2C6;&#x2019;6

Zeros: 1 of multiplicity 2; 3 of multiplicity 1;

9 2

â&#x2C6;&#x2019;5

multiplicity 1 92. h x

1 5

2 3x 5

â&#x2C6;&#x2019;10

The function has two zeros.

They are in the intervals > 2, 1@ and >0, 1@. They are x | 1.585, 0.779.

â&#x2C6;&#x2019; 12

12 â&#x2C6;&#x2019;3

Zeros: 2, 93. f x

5 , 3

509

132

348

both with multiplicity 2

x 3x 3 3

96. h x

x 4 10 x 2 3

â&#x2C6;&#x2019;5

â&#x2C6;&#x2019;10

The function has three zeros. They are in the intervals > 1, 0@, >1, 2@ , and >2, 3@. They are x | 0.879, 1.347, 2.532.

The function has four zeros. They are in the intervals > 4, 3@, > 1, 0@, >0, 1@, and >3, 4@.

They are approximately r 3.113 and r0.556. 10 â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;30

INSTRUCTOR CTO USE ONLY 3

NOT FOR SALE Section 2.2

l Â&#x2DC;wÂ&#x2DC;h 97. (a) Volume x height length width 36 2 x So, V x

Polynomial Functions of Higher Hi H Degree

99. (a) A

36 2 x 36 2 x x

x 36 2 x .

2 x 2 12 x

l Â&#x2DC;wÂ&#x2DC;h

2 x x 192

384 x 2 2304 x (c) Because x and 12 2x cannot be negative, we have 0 x 6 inches for the domain.

The length and width must be positive. Box Height

Box Width

Box Volume, V

36 2 1

1ÂŞÂŹ36 2 1 ÂşÂź

36 2 2

2 ÂŞÂŹ36 2 2 ÂşÂź

36 2 3

3ÂŞÂŹ36 2 3 ÂşÂź

36 2 4

4 ÂŞÂŹ36 2 4 ÂşÂź

36 2 5

5ÂŞÂŹ36 2 5 ÂşÂź

36 2 6

6 ÂŞÂŹ36 2 6 ÂşÂź

36 2 7

7 ÂŞÂŹ36 2 7 ÂşÂź

(d)

1156 2

2 2

1920

3072

3136

3456

3380

3072

3456

1920

3388

2048 2700

The volume is a maximum of 3456 cubic inches when the height is 6 inches and the length and width are each 24 inches. So the dimensions are 6 u 24 u 24 inches. (d)

2 x x

(b) 16 feet = 192 inches

(b) Domain: 0 x 18 (c)

l Â&#x2DC;w

147

When x

3, the volume is a maximum with

V 3456 in.3. The dimensions of the gutter crosssection are 3 inches u 6 inches u 3 inches. (e)

4000

3600

0 0

18 0

The maximum point on the graph occurs at x This agrees with the maximum found in part (c). 98. (a) Volume

(b) x ! 0,

lÂ&#x2DC;wÂ&#x2DC;h

24 2 x 24 4 x x 2 12 x Â&#x2DC; 4 6 x x 8 x 12 x 6 x

12 x ! 0,

6 x ! 0

x 12

x 6

Maximum: (3, 3456)

The maximum value is the same. (f ) No. The volume is a product of the constant length and the cross-sectional area. The value of x would remain the same; only the value of V would change if the length was changed. 100. (a) V

4S r 3 3

S r 2 4r

4S r 3 3

4S r 3

16 S r3 3

Domain: 0 x 6

(b) r t 0

(c)

720

150

600 480 360 0

240

120 x 1

(d) V

x | 2.5 corresponds to a maximum of 665 cubic inches.

120 ft 3

16 S r3 3

r | 1.93 ft length

4r | 7.72 ft

148

Chapter 2

101. (a)

NOT FOR SALE

Polynomial ynomial and Rational Function Functions

1000

104. G

0.003t 3 0.137t 2 0.458t 0.839, 2 d t d 34

(a) 3

â&#x2C6;&#x2019; 50

Relative maximum: 5.01, 655.75

Relative minimum: 9.25, 417.42

(b) The revenue is increasing over 3, 5.01 and decreasing over 5.01, 9.25 , and then increasing over 9.25, 10 .

0.009t 2 0.274t 0.458

b 2a

0.274 | 15.222 2 0.009

Vertex | 15.22, 2.54

(d) The x-value of the vertex in part (c) is approximately equal to the value found in part (b).

106. False. f has at least one real zero between x x 6.

â&#x2C6;&#x2019; 20

Relative maxima: 4.11, 21.87 , 9.02, 29.25

Relative minimum: 6.01, 18.81

(b) The revenue was increasing over 3, 4.11 and

6.01, 9.02 and was decreasing over 4.11, 6.01

and 9.02, 10 . (c) The revenue for this company was increasing from 2003 to 2004 when it reached a (relative) maximum of $21.87 million. From 2004 to 2006 when revenue dropped to a (relative) minimum of $18.81 million. From 2006 to 2009, revenue again was increasing to a (relative) maxima of $29.25 million. From 2009 to 2010, revenue again began to decrease. 103. R

(b) The tree is growing most rapidly at t | 15.

105. False. A fifth-degree polynomial can have at most four turning points.

45 â&#x2C6;&#x2019;5

y 15.222 | 2.543

(c) The revenue for this company is increasing from 2003 to 2005, when it reached a (relative) maximum of $655.75 million. From 2005 to 2009, revenue was decreasing when it dropped to $417.42 million. From 2009 to 2010, revenue began to increase again. 102. (a)

â&#x2C6;&#x2019; 10

1 x3 600 x 2

100,000

107. False. The function f x

2 and

2 has one turning 2

point and two real (repeated) zeros. 108. True. A polynomial function only falls to the right when the leading coefficient is negative. 109. False. f x

x3 rises to the left.

110. False. The graph rises to the left and to the right. 111. True. A polynomial of degree 7 with a negative leading coefficient rises to the left and falls to the right. 112. (a) Degree: 3

Leading coefficient: Positive (b) Degree: 2 Leading coefficient: Positive (c) Degree: 4

The point of diminishing returns (where the graph changes from curving upward to curving downward) occurs when x 200. The point is 200, 160 which

Leading coefficient: Positive (d) Degree: 5 Leading coefficient: Positive

corresponds to spending $2,000,000 on advertising to obtain a revenue of $160 million.

NOT FOR SALE Section 2.2

Polynomial Functions of Hi Higher Degree H

149

113. Answers will vary. Sample answers:

a4 0

a4 ! 0 y

4 3 2

1 x

â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;4

114. Answers will vary. Sample answers:

a5 0

a5 ! 0 y

10 8 6 4

â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019; 10

115. f x

x 4 ; f x is even.

(a) g x

f x 2

4 3

Vertical shift two units upward g x

f x 2

2 1

f x 2

x â&#x2C6;&#x2019;3

g x

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;1

Even (b) g x

f x 2

(c)

Horizontal shift two units to the left Neither odd nor even (d) g x

f x

1 2

f x

(e)

x 4

g x

12 x

1 x4 16

Horizontal stretch Even

1 x4 2

(g)

Vertical shrink Even (h) g x

f x

Reflection in the y-axis. The graph looks the same. Even

Reflection in the x-axis Even (f ) g x

g x

f x3 / 4

x3 / 4

x3 , x t 0

Neither odd nor even f f x

D f x

f x4

x16

Even 116. (a) f x

x3 2 x 2 x 1

3, odd; 1, positive

(b)

f x

(c)

2 x 5 x 2 5 x 3

5, odd; 2, negative

5, odd; 2, positive

â&#x2C6;&#x2019;4

2 x5 2 x 2 5 x 1

INSTRUCTOR USE ONLY 5

â&#x2C6;&#x2019;7

â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;7

â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;3

3 â&#x2C6;&#x2019;3

150

NOT FOR SALE

Chapter 2 (d) f x

Polynomial ynomial and Rational Function Functions x3 5 x 2

(e)

3, odd; 1, negative

f x

(f )

2, even; 2, positive

x 4 3x 2 2 x 1

4, even; 1, positive

â&#x2C6;&#x2019;7

â&#x2C6;&#x2019;8

â&#x2C6;&#x2019;7

(g) f x

2 x2 3x 4

â&#x2C6;&#x2019;7

x 3x 2 2

2, even; 1, positive 5

â&#x2C6;&#x2019;5

4 â&#x2C6;&#x2019;1

When the degree of the function is odd and the leading coefficient is positive, the graph falls to the left and rises to the right, but if the leading coefficient is negative, the graph falls to the right and rises to the left. When the degree of the function is even and the leading coefficient is positive, the graph rises to the left and right, but if the leading coefficient is negative, the graph falls to the left and right. y

117. (a)

(c)

9 6 3 â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;5

x 1

â&#x2C6;&#x2019;10

â&#x2C6;&#x2019;9

â&#x2C6;&#x2019;15

â&#x2C6;&#x2019;12

â&#x2C6;&#x2019;20

Zeros: 3

Relative minimum: 1

Relative maximum: 1

The number of zeros is the same as the degree and the number of extrema is one less than the degree.

The number of zeros and the number of extrema are both less than the degree.

(b) 16 12

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;2

x â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;8 â&#x2C6;&#x2019;12 â&#x2C6;&#x2019;16

Zeros: 4 Relative minima: 2 Relative maximum: 1 The number of zeros is the same as the degree and the number of extrema is one less than the degree.

NOT FOR SALE Section 2.3

13 x 2 1 is decreasing. 5

118. (a) y1

3 5

Polynomi Polynomial al and Synthetic Synth Division

151

x5 3x3 2 x 1

Because H x is not always increasing or always

2 3 is increasing. 5

decreasing, H x cannot be written in the form

a x h k . 5

â&#x2C6;&#x2019; 12

y2 â&#x2C6;&#x2019;8

â&#x2C6;&#x2019;9

(b) The graph is either always increasing or always decreasing. The behavior is determined by a. If a ! 0, g x will always be increasing. If

â&#x2C6;&#x2019;6

a 0, g x will always be decreasing.

Section 2.3 Polynomial and Synthetic Division 1. f x is the dividend; d x is the divisor: q x is the

quotient: r x is the remainder

9. y1

x2 2 x 1 , y2 x 3 3

(a) and (b)

2. proper

2 x 3

x 1

â&#x2C6;&#x2019;9

3. improper 4. synthetic division

â&#x2C6;&#x2019;9

5. Factor

x 1 (c) x 3 x 2 2 x 1 x2 3x x 1 x 3 2

6. Remainder 7. y1

x2 and y2 x 2

x 2

4 x 2

x 2 x 2 x2 0x 0 x2 2x 2 x 0 2 x 4 4

So,

8. y1

x2 x 2

10. y1 4 and y1 x 2

x 2

x 4 3x 2 1 and y2 x2 5

x2 2 4 x 5 x 3x 2 x4 5x2 8 x 2 8 x 2 x 3x 1 So, x2 5 4

x2 2x 1 x 3

x 1

x4 x2 1 , y2 x2 1

So,

x2 8

(a) and (b)

y2 . 39 x2 5

2 and y1 x 3

y2 .

1 x2 1

â&#x2C6;&#x2019;6

6 â&#x2C6;&#x2019;2

8 1

1 40 39

So,

39 and y1 x 8 2 x 5 2

x4 x2 1 x2 1

1 and y1 x2 1

y2 .

y 2.

152

Chapter 2

NOT FOR SALE

Polynomial ynomial and Rational Function Functions 2x 4

x 2 7 x 18

11. x 3 2 x 10 x 12

16. x 3 x 4 x 2

2x 6x 4 x 12 4 x 12 0

2 x 2 10 x 12 2 x 4, x z 3 x 3 5x 3

12. x 4 5 x 17 x 12 5 x 2 20 x 3 x 12 3 x 12 0

3x 1

4 x3 7 x 2 11x 5 4x 5

x3 27 x 3

x 2 5 x 25

x 2 3x 1, x z

18. x 5 x 0 x 2 0 x 125 x3 5 x 2 5 x 2 0 x 5 x 2 25 x 25 x 125 25 x 125 0

5 4

14. 3x 2 6 x3 16 x 2 17 x 6

x3 125 x 5

6 x3 4 x 2 12 x 2 17 x 12 x 8 x 9x 6 9x 6

7x 3 x 2

0 2 x 2 4 x 3, x z

2 3

1 x3 3x 2 15. x 2 x 5 x3 6 x 2 x 2 4

11 x 2

4 20. 2 x 1 8 x 5 8x 4 9 8x 5 2x 1

9 2x 1

x 21. x 2 0 x 1 x3 0 x 2 0 x 9 x3 0 x 2 x x 9

x 2 x 2 0

x 4 5 x3 6 x 2 x 2 x 2

x 2 5 x 25, x z 5

7 19. x 2 7 x 3 7 x 14 11

6 x3 16 x 2 17 x 6 3x 2

x 2 3 x 9, x z 3

2x2 4 x 3

x 4 2 x3 3x3 6 x 2 3x3 6 x 2

42 x 3

17. x 3 x3 0 x 2 0 x 27 x3 3x 2 3x 2 0 x 3x 2 9 x 9 x 27 9 x 27 0

13. 4 x 5 4 x 7 x 11x 5 4 x3 5 x 2 12 x 2 11x 12 x 2 15 x 4x 5 4x 5 0 3

x 2 7 x 18

x 2 3x 9

5 x 3, x z 4

x3 4 x 2 3 x 12 x 3

5 x 2 17 x 12 x 4

3 x 12

x 3x 7 x 2 3x 7 x 2 21x 18 x 12 18 x 54 42

x3 3x 2 1, x z 2

x3 9 x2 1

x 9 x2 1

NOT FOR SALE Section 2.3

x2 22. x 0 x 0 x 1 x 0 x 0 x 0 x 2 0 x 7 x5 0 x 4 0 x3 x 2 7 x2 3

x 7 x3 1

28.

29.

2x 8 3x 9 2x x 9 0x 8 x 1

0x2 3x 2 3x 2 6x2 9 x2 9 x2

x 1

x 1 x2 1

30.

20 x 18 x 2 x 16 9 x 27 7 x 11

0x 3x 2 3x 2 9 x2 6x2

x 3 0x 0 x x 0 9x 3 8x 3

6x2 8x 3

x 1

222

248 6 x 2 25 x 74

192

199

2 x 14 x 20 x 7 x 6

20 x 16

31.

2 x 2 2 x 32

4 x3 8 x 2 9 x 18 x 2

32.

9 x3 18 x 2 16 x 32 x 2

33.

1 1

248 x 3

199 x 6

4 x 2 9, x z 2

153

5 x 2 3 x 2, x z 3

7 x 11 x 6x 9 2 x x 3

x 3

6 x 7 x x 26 x 3

25. x 3x 3 x 1 x 0 x x 4 3x3 3x3 3x3 3

x2 6x 9

x 4 5 x3 20 x 16 x2 x 3

2x 8

24. x 2 x 3 x 4 5 x3 x 4 x3 6 x3 6 x3

5 x3 18 x 2 7 x 6 x 3

2 x3 8 x 2 3x 9 x2 1

5 5

23. x 2 0 x 1 2 x3 8 x 2 2 x3 0 x 2 8 x 2 8 x 2

x 7 x 3 x 1

Polynomi Polynomial al and Synthetic Synth Division

9 x 2 16, x z 2

250

100

250

2x x3 75 x 250 x 10

26. x 2 2 x 1 2 x3 4 x 2 15 x 5 2 x3 4 x 2 2 x 17 x 5

2 x3 4 x 2 15 x 5

x 27.

3 3

17 x 5 2x 2 x 2x 1

3x 17 x 15 x 25 x 5 3

34.

x 2 10 x 25, x z 10

3x 16 x 72 x 6 3

35.

3x 2 x 5, x z 5 2

3x 2 2 x 12, x z 6

224

232

5 x3 6 x 2 8 x 4

232 x 4

INSTRUCTOR USE ONLY 5 x 2 14 x 56

154 36.

NOT FOR SALE

Chapter 2

Polynomial ynomial and Rational Function Functions

5 5

5 x3 6 x 8 x 2

37.

800

360

2160

360

1360

10 x 50 x 800 x 6 4

38.

1 1

10 x3 10 x 2 60 x 360

120

144

432

936

144

312

856

x 13 x 120 x 80 x 3 5

39.

1 1

512

x 512 x 8

3 3 x x 2

42.

3 x 4 x 2

729

45.

3 x3 6 x 2 12 x 24

1 180 x x 4 x 6

â&#x20AC;&#x201C;6

5 3x 2 x x x 1

1 2

x 2 3x 6

4 x3 16 x 2 23 x 15 1 x 2 46.

3 2

48 3 x3 6 x 2 12 x 24 x 2

43.

x 2 9 x 81, x z 9

44.

x3 729 x 9

856 x 3

x 2 8 x 64, x z 8

41.

1360 x 6

x 4 16 x3 48 x 2 144 x 312

40.

44 x 2

5 x 2 10 x 26

48 x 2

180

216

11 x 1

4 x 2 14 x 30, x z

9 2 1 2

3 4 3 4

9 8 49 8

3x 2

1 3 49 x 2 4 8 x 12

3x3 4 x 2 5 3 x 2

1 2

216 x 6

INSTRUCTOR USE ONLY x3 6 x 2 36 x 36

NOT FOR SALE Section 2.3

47. f x

x3 x 2 14 x 11, k

50. f x

1 5

f 4

43 42 14 4 11

48. f x

x3 5 x 2 11x 8, k

13 5

1 1

f x

f 2

49. f x

51. f x

2 x 2 7 x 3 2 3

5 2 11 2 8

34 3

52. f x

1 1

f x

53. f x

3 3

2 3 2

3 2

2Âº 8 Â¼ 8

2 5 5

2 5

5 x 2 5Âº 6 Â¼

4 x3 6 x 2 12 x 4, k 3

34 3

5 Âªx 2 2 Â¬

5 4

3 12

4 4 3

10 2 3

2 4 3

2 2 3

x 1 3 ÂªÂ¬ 4 x 2 4 3 x 2 2 3 ÂºÂ¼ 4 1 3 6 1 3 12 1 3 4 3

f x

f 1

x 2 Â¬Âªx 3 2 x 3 f 2 2 3 2 2 2 14

x3 2 x 2 5 x 4, k

x3 3 x 2 2 x 14, k

f x

x 23 15x 6 x 4 343 f 23 15 23 10 23 6 23 14 f x

1 5

15 x 4 10 x3 6 x 2 14, k

155

x 15 10 x 20 x 7 135 13 f 15 10 15 22 15 3 15 4 5 f x

â&#x20AC;&#x201C;2

10 x3 22 x 2 3x 4, k

4 x 2 3 x 2 3

f x

Polynomial and Synth Synthetic Division

156

NOT FOR SALE

Chapter 2

54. f x

Polynomial ynomial and Rational Functions

3 x3 8 x 2 10 x 8, k

2 10

6 3 2

2 4 2

2 3 2

8 4 2

x 2 2 ÂŞÂŹ 3x 2 3 2 x 8 4 2 ÂşÂź 2

3 2 2 8 2 2 10 2 2 8

f x

f 2

55. f x

2 x3 7 x 3

(a) Using the Remainder Theorem: f 1

2 1 7 1 3 3

(b) Using the Remainder Theorem: f 2

Using synthetic division: 1

2x x 1 2x 0 x2 2 x3 2 x 2 2x2 2x2 3

â&#x20AC;&#x201C;2

2x 5 7x 3 7x 2x 5x 3 5x 5 2

1 2

1 2 13 2

1 4

f 2

â&#x20AC;&#x201C;2

2 2 7 2 3 3

13 4 1 4

0 4

4x 1 7x 3

Using synthetic division: 2

â&#x20AC;&#x201C;4 â&#x20AC;&#x201C;4

7x 8x x 3 x 2 1 (d) Using the Remainder Theorem:

2x2 x 2 2x 0 x2 2 x3 4 x 2 4x2 4x2

Â§1Âˇ Â§1Âˇ 2Â¨ Â¸ 7Â¨ Â¸ 3 2 ÂŠ Âš ÂŠ 2Âš

Using synthetic division: 1 2

Verify using long division:

Using synthetic division:

Verify using long division: 2

2 2 7 2 3

Verify using long division: 2x2 4 x 1 x 2 2x 0 x2 7 x 3 2 x3 4 x 2 4x2 7 x 4x2 8x x 3 x 2 5 3

Verify using long division: 13 2 1 x 2 x3 0 x 2 7 x 3 2 2 x3 x 2 2x2 x

x2 7 x 1 x2 x 2 13 x 3 2 13 13 x 2 4 1 4

NOT FOR SALE Section 2.3

56. g x

Polynomi Polynomial al and Synth Synthetic Division

157

2 x6 3x 4 x2 3

(a) Using the Remainder Theorem: g 2

(b) Using the Remainder Theorem:

2 2 3 2 2 3 6

g 1

175

Using synthetic division: 2

2 2

172

175

g 3

4 x 11x 22 x 43x 86 3 x 4 0 x3 x 2 0 x 3 3

3x 4 8x4 11x 4 0 x3 11x 4 22 x3 22 x3 x 2 22 x3 44 x 2 43 x 2 0 x 43 x 2 86 x 86 x 3 86 x 172 175

2 2

189

564

1692

188

564

1695

2 x 4 5 x3 5 x 2 4 x 4 3x 4 0 x3 x 2 0 x 3 3x 4 2x4 5 x 4 0 x3 5 x 4 5 x3 5 x3 x 2 5 x3 5 x 2 4x2 0x 4x2 4x 4x 3 4x 4 7

2 1 3 1 1 3 6

â&#x20AC;&#x201C;1

â&#x20AC;&#x201C;4

2 2

Verify using long division:

6 x 21x 63 x 188 x 564 4

x 3 2 x 0 x 3x 0 x x 0x 3 2 x 6 6 x5 6 x5 3 x 4 6 x 4 18 x 4 21x 4 0 x3 21x 4 63 x3 x2 63x3 3 63x 189 x 2 188 x 2 0 x 188 x 2 564 x 564 x 3 564 x 1692 1695 5

Using synthetic division:

Verify using long division: 6

2 x5 x 1 2 x 6 0 x5 2 x 6 2 x5 2 x5 2 x5

g 1

1695

(d) Using the Remainder Theorem:

2 3 3 3 3 3 4

Using synthetic division: 3

Verify using long division:

2 2

Verify using long division: 2x x 2 2 x6 0 x5 2 x6 4 x5 4 x5 4 x5

Using synthetic division:

2 1 3 1 1 3

2 x5 x 1 2 x 0 x5 2 x6 2 x5 2 x5 2 x5 6

2 x 4 5 x3 5 x 2 4 x 4 3 x 4 0 x3 x 2 0 x 3 3x 4 2x4 5 x 4 0 x3 5 x 4 5 x3 5 x3 x 2 5 x3 5 x 2 4x2 0x 4x2 4x 4 x 3 4 x 4 7

158

NOT FOR SALE

Chapter 2

57. h x

Polynomial ynomial and Rational Functions

x3 5 x 2 7 x 4

(a) Using the Remainder Theorem: h 3

(b) Using the Remainder Theorem:

5 3 7 3 4

h 2

Using synthetic division: 3

1 1

x x 3 x 5x2 x3 3x 2 2 x 2 2 x 2

h 2

2 x 13 7x 4

x2 x 2 x 5x2 x3 2 x 2 3x 2 3x 2 3

7x 6x 13 x 4 13 x 39 35

5 2 7 2 4

3 x 13 7x 4 7x 6x 13 x 4 13 x 26 22

h 5

5 3

5 5 7 5 4 2

211

Using synthetic division:

â&#x20AC;&#x201C;5

Verify using long division:

215

211

Verify using long division:

x2 7 x 7

x 2 10 x 43

x 2 x3 5 x 2 7 x 4 x3 2 x 2 7 x 2 7 x 7 x 2 14 x 7x 4 7 x 14 10

x 5 x3 5 x 2 7 x 4 x3 5 x 2 10 x 2 7 x 10 x 2 50 x 43 x 4 43 x 215 211

58. f x

(d) Using the Remainder Theorem:

Using synthetic division: â&#x20AC;&#x201C;2

Verify using long division:

1 1

Verify using long division: 3

5 2 7 2 4

Using synthetic division:

2 3

4 x 4 16 x3 7 x 2 20

(a) Using the Remainder Theorem: f 1

(b) Using the Remainder Theorem:

4 1 16 1 7 1 20 4

Using synthetic division: 1

4 4

f 2

4 2 16 2 7 2 20 4

240

Using synthetic division:

20 5

Verify using long division: 4 x 3 12 x 2 5 x x 1 4 x 4 16 x 3 7 x 2 0 x 4 x 4 4 x3 12 x 3 7 x 2 12 x 3 12 x 2 5 x 2 0 x 5 x 2 5 x 5 x 5 x

5 20

20 5 15

4 4

110

220 240

Verify using long division: 4 x3 24 x 2 55 x 110 x 2 4 x 4 16 x3 7 x 2 0 x 20 4 x3 8 x3 24 x3 7 x 2 24 x3 48 x 2 55 x 2 0 x 55 x 2 110 x 110 x 20 110 x 220 24 240

NOT FOR SALE Section 2.3

695

Using synthetic division: 5

135

675

135

695

1 1

x x

x3 7 x 6

560

5670

567

5670

56,700 56,720

4 x 27 x 135

4 x3

56 x 2 567 x

62.

2 x 2 2 x 3

2 3

x 23 48 x 48x 9

x 23 4 x 3 12 x 3

48 x3 80 x 2 41x 6

2 x 3 x 1

3 x

x3 28 x 48

x x

Zeros: 23 , 34 , 63.

1 2

2 2

4 x 2 4 x 12

3 2 3

2 3

2 x3 15 x 2 27 x 10

x 12 2 x 2 x

1 x 2 x 5

64.

Zeros: 12 , 2, 5

3 x

2 2 2

2 2

x 2x 2x 4 2

Zeros: 2,

3 x 2

1 3

2 3

1 2

3, 2

2 3

Zeros: 3,

14 x 20

x 2 x 3x 6 3

2 4 x 3 4 x 1

1 4

4 x 6 x 2

Zeros: 4, 2, 6 61.

5670

7x 0x 20 x 10 4 x 16 x 4 x 4 40 x3 56 x3 7 x 2 56 x3 560 x 2 567 x 2 0x 567 x 2 5670 x 20 5670 x 5670 x 56,700 56,720 4

Zeros: 2, 3, 1 60.

56,720

Verify using long division:

x 5 4 x 4 16 x3 7 x 2 0 x 20 4 x 4 20 x3 4 x3 7 x 2 4 x 3 20 x 2 27 x 2 0 x 27 x 2 135 x 135 x 20 135 x 675 695 59.

4 10 16 10 7 10 20

Verify using long division: 3

f 10

Using synthetic division:

159

(d) Using the Remainder Theorem:

4 5 16 5 7 5 20 4

Polynomial and Synthetic Synth Division

2 x 2 x

INSTRUCTOR USE ONLY 2,

160 65.

NOT FOR SALE

Chapter 2 1

Polynomial ynomial and Rational Function Functions 3

1 1

â&#x20AC;&#x201C;1

3x3 2 x 2 19 x 6;

Factors: x 3 , x 2

(a)

3 2

3 ÂźÂş x 1

3 x 1 3

3 ÂşÂŞ x 1 ÂźÂŹ

x 1 x 1 3, 1

68. f x

ÂŞx 1 ÂŹ

Zeros: 1, 1

1 x3 3x 2 2

Both are factors of f x because the remainders

are zero. 66.

7 3 5

6 3 5

1 x 3 x 2 13 x 3

Zeros: 2

5, 2

5 x 2

2 1

2 2

3 x

5 x 3

â&#x2C6;&#x2019;4

3 â&#x2C6;&#x2019;10

69. f x

x 4 4 x3 15 x 2 58 x 40;

Factors: x 5 , x 4

(a)

(d) Zeros:

1 x 2 x 1

40 0

Both are factors of f x because the remainders are zero. (b) x 2 3 x 2

1 x 2

The remaining factors are x 1 and x 2 .

2, 1

(e)

1 1

are zero.

1 , 2

1 1

(b) The remaining factor of f x is 2 x 1 .

2 x

1 x 3 x 2

(e)

Both are factors of f x because the remainders

3x3 2 x 2 19 x 6

5, 3

(d) Zeros: 13 , 3, 2

2 x3 x 2 5 x 2; Factors: x 2 , x 1

67. f x

(a)

x 2

(b) The remaining factor is 3x 1 .

1 x 2 x 5 x 4

(d) Zeros: 1, 2, 5, 4 â&#x2C6;&#x2019;6

(e)

20 â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;1

â&#x2C6;&#x2019; 180

NOT FOR SALE Section 2.3

70. f x

71. f x

8 x 4 14 x 3 71x 2 10 x 24;

Factors: x 2 , x 4

(a)

8 8

24 24

6 x3 41x 2 9 x 14;

(a)

6 6

2 3

Both are factors of f x because the remainders

are zero.

4 x

(b) 8 x 2 2 x 3

3 2 x 1

f x

(c)

4 x

This shows that

3 2 x 1 x 2 x 4

(d) Zeros: 34 , 12 , 2, 4

40 â&#x2C6;&#x2019;3

6 x 7

(b) 6 x 42

The remaining factors are 4 x 3 and 2 x 1 .

(e)

161

Factors: 2 x 1 , 3x 2

Polynomial and Synth Synthetic Division

2 x

f x

1 Â·Â§ 2Â· Â§ Â¨ x Â¸Â¨ x Â¸ 2 3Â¹ Â© Â¹Â©

f x

1 3 x 2

6 x 7 ,

x 7.

The remaining factor is x 7 .

(c)

f x

7 2 x 1 3 x 2

1 2 (d) Zeros: 7, , 2 3

â&#x2C6;&#x2019; 380

320

(e)

â&#x2C6;&#x2019;9

3 â&#x2C6;&#x2019; 40

72. f x

10 x3 11x 2 72 x 45;

Factors: 2 x 5 , 5 x 3

(a)

10 10

3 5

10 10

(b) 10 x 30

10 x 3

This shows that

2 x

f x

5 3Â· Â§ Â·Â§ Â¨ x Â¸Â¨ x Â¸ 2 Â¹Â© 5Â¹ Â©

f x

5 5 x 3

10 x 3 ,

x 3.

The remaining factor is x 3 .

Both are factors of f x because the remainders are zero. (c)

f x

3 2 x 5 5 x 3

5 3 (d) Zeros: 3, , 2 5

100

(e) â&#x2C6;&#x2019;4

INSTRUCTOR USE ONLY â&#x2C6;&#x2019; 80

162

Chapter 2

73. f x

NOT FOR SALE

Polynomial ynomial and Rational Functions

75. f x

2 x3 x 2 10 x 5;

Factors: 2 x 1 , x

(a)

1 2

(b) An exact zero is x (c)

2 5

(b) 2 x 2 5

This shows that

2 x

(d) Zeros: 5,

2 x 5

2 x

(c)

77. h t

1 2

4 x 2 2

4 x

â&#x2C6;&#x2019;6

74. f x

x 3x 48 x 144; 2

h t

Factors: x 4 3 , x 3

(a)

144

1 4 3

4 3

1 1

t 3 2t 2 7t 2 2, t | 3.732, t | 0.268.

(b) An exact zero is t â&#x2C6;&#x2019;6

2 x

(a) The zeros of h are t

(c)

4, x | 1.414, x | 1.414.

4 is an exact zero.

5 .

5 x

(b) x

g x

(e)

5 2 x 1

5 x

x3 4 x 2 2 x 8

The remaining factor is x (c) f x

76. g x

(a) The zeros of g are x

1 x

f x

1Âˇ Â¸ x 2Âš

Â§ Â¨x ÂŠ

f x

Both are factors of f x because the remainders are zero.

2 and x | r2.236.

(a) The zeros of f are x

2 5

x3 2 x 2 5 x 10

2 t 2 4t 1

By the Quadratic Formula, the zeros of t 2 4t 1 are 2 r 3. Thus, h t

2 ÂŞt 2 ÂŹ

3 ÂşÂŞt 2 ÂźÂŹ

3 Âş. Âź

Both are factors of f x because the remainders are zero.

(b) The remaining factor is x 4 3 . (c) f x

x 4 3 x 4 3 x 3

(d) Zeros: r 4 3, 3 60

(e) â&#x2C6;&#x2019;8

INSTRUCTOR USE ONLY â&#x2C6;&#x2019; 240

NOT FOR SALE Section 2.3

78. f s

s 3 12 s 2 40 s 24

81.

6, s | 0.764, s | 5.236

(a) The zeros of f are s (b) s

6 is an exact zero.

(c)

f s

6 s 6s 4

6 Âªs 3 Â¬

5 ÂºÂªs 3 Â¼Â¬

5Âº Â¼

x 5 7 x 4 10 x3 14 x 2 24 x

(a) The zeros of h are x

0, x

3, x

82.

x | 1.414, x | 1.414.

(b) An exact zero is x (c)

1 1

h x

x x 4 x 3 x

80. g x

2 x

6 x 4 11x3 51x 2 99 x 27

(a) The zeros of a are x

3, x

83.

a x

84.

x x

1 1

108

27 0

3 6 x3 29 x 2 36 x 9

3 x 3 2 x 3 3 x 1

x 4 9 x3 5 x 2 36 x 4 x2 4

2 2 x 2 x 1

2 x 2 x 1, x z

3 . 2

0 x 2 7 x 8, x z 8

1 1

x 4 6 x3 11x 2 6 x x 1 x 2

x 2 3 x, x z 2, 1

x 4 9 x3 5 x 2 36 x 4 x 2 x 2

4x2 2 x 2

1.5,

x 4 9 x 3 5 x 2 36 x 4 x2 4

(b) An exact zero is x â&#x20AC;&#x201C;3

x 4 6 x 3 11x 2 6 x x 2 3x 2

x | 0.333. (c)

x x 64 x 64 x 8

4 x3 8 x 2 x 3 2x 3

1 3

4 x 3 x 2 x 6 x

x3 x 2 64 x 64 x 8

4 x3 8 x 2 x 3 3 x 2 So,

79. h x

163

4 x3 8 x 2 x 3 2x 3

3 2

Polynomial and Synthetic Synth Division

x 2 9 x 1, x z r2

164

NOT FOR SALE

Chapter 2

85. (a)

Polynomial ynomial and Rational Functions

3,200,000

â&#x2C6;&#x2019; 400,000

(b) Using the trace and zoom features, when x the same profit of $2,174,375.

25, an advertising expense of about $250,000 would produce

25 152

152

7545

169,625

3800

93,625

2,340,625

3745

93,625

2,171,000

So, an advertising expense of $250,000 yields a profit of $2,171,000, which is close to $2,174,375. 86. (a) and (b)

1000

15 0

3.1705t 4 71.205t 3 551.75t 2 1821.2t 1985

(c)

179

217

246

351

539

743

821

552

The estimated values are close to the original data values. (d)

3.1705

551.75

1821.2

1985

31.705

395 156.75

1567.5 253.7

2537

39.5

71.205

552, you can conclude that N 10

Because the remainder is r

87. False. If 7 x 4 is a factor of f , then 74 is a zero

of f . 88. True. 1 2

6 6

f x

2 x

184

1 x 1 x 2 x 3 3 x 2 x 4

89. True. The degree of the numerator is greater than the degree of the denominator. x3 3x 2 4 90. False. The equation x 2 4 x 4 it is x 1 not true for x 1 since this value would result in

division by zero in the original equation. So, the equation x3 3x 2 4 x 2 4 x 4, should be written as x 1 x z 1.

552

552. This confirms the estimated value. x2n 6 xn 9 91. x 3 x 9 x 2 n 27 x n 27 x3n 3 x 2 n 6 x 2 n 27 x n 6 x 2 n 18 x n 9 x n 27 9 x n 27 0 n

x 3n 9 x 2 n 27 x n 27 xn 3

x 2 n 6 x n 9, x n z 3

x 2n x n 3 92. x n 2 x3n 3 x 2 n 5 x n 6 x 3n 2 x 2 n x2n 5xn x2n 2 xn 3x n 6 3x n 6 0 x 3n 3 x 2 n 5 x n 6 xn 2

INSTRUCTOR USE ONLY x 2 n x n 3,, x n z 2

NOT FOR SALE Section 2.4

Comp Complex Numbers

165

Section 2.4 Complex Numbers 13. 2

1. real

2 3 3i

2. imaginary 14. 1

3. pure imaginary

1; 1

5. principal square

12 7i

15.

16.

6. complex conjugates 7. a bi

27i

1 2 2i 4 5i

17. 14

14 0i

75 0i

18. 75

19. 10i i 2

10i 1

20. 4i 2 2i

4 1 2i

8. a bi

13 4i

4 2i

9. a 1 b 3 i

5 8i

a 1

5 Â&#x; a

b 3

8 Â&#x; b

10. a 6 2bi

a 6 2b

1 10i

21.

5 Â&#x; b

22.

8 5i

12. 5

5 6i

0.0049

0.0049i 0.07i

23. 7 i 3 4i

11. 8

0.09i 0.3i

6 5i

6 Â&#x; a

0.09

10 3i

24. 13 2i 5 6i

25. 9 i 8 i

8 4i

26. 3 2i 6 13i

3 2i 6 13i 3 11i

27. 2

8 5

2 2 2i 5 5 2i 3 3 2i

28. 8

18 4 3 2i

8 3 2i 4 3 2i

31.

23 52 i 53 113 i

29. 13i 14 7i

5 3

i 96 15 6

10 6

1 6

13i 14 7i 14 20i

30. 25 10 11i 15i

23 52 i

11 i 3

22 i 6

76 i

32. 1.6 3.2i 5.8 4.3i

4.2 7.5i

15 26i

166

NOT FOR SALE

Chapter 2

Polynomial ynomial and Rational Functions

33. 1 i 3 2i

45. The complex conjugate of 1

3 2i 3i 2i 2 3 i 2

34. 7 2i 3 5i

5 i

1 5

12i 108i 2

2i 3

10i

2 5i 2 5i

10i

14 10i 2 14 10

38.

15i

6 6

15i 2

2 5i is 2 5i .

15i is 15i.

15 6 is

50. The complex conjugate of 1

8 1

51.

4 12i 9i 2 4 12i 9i 2 4 12i 9 4 12i 9

42. 1 2i 1 2i

1 4i 4i 1 4i 4i

1 4i 4i 1 4i 4i 2

53.

8i 43. The complex conjugate of 9 2i is 9 2i. 2i 9 2i

3i i2

14 2i Â&#x2DC; 2i 2i

2 4 5i

3i 28i 4i 2

13 1 i

Â&#x2DC; 1 i 1 i

55.

5 i 5 i

Â&#x2DC; 5 i 5 i

64 100i 2 64 100

13 13i 1 i2

56.

6 7i 1 2i Â&#x2DC; 1 2i 1 2i

164

8 10 i 41 41

13 13i 2

13 13 i 2 2

25 10i i 2 25 i 2 24 10i 26

44. The complex conjugate of 8 10i is 8 10i.

8 10i 8 10i

28i 4

2 4 5i Â&#x2DC; 4 5i 4 5i 2 4 5i

8 10i 16 25 41

54.

81 4i 2 81 4

3 i Â&#x2DC; i i

52.

10 2

9 4 2

9 40i

41. 2 3i 2 3i

8 is 1

1 2 8 8

25 40i 16i 2 25 40i 16

36 84i 49i 2 13 84i

15i 15i

49. The complex conjugate of 18

36 84i 49

40. 5 4i

20i 2

48. The complex conjugate of

3 15i 2 3 15

39. 6 7i

9 2i 2

47. The complex conjugate of

72i 32i 2 32 72i

2i.

108 12i

37.

2i is 3

9 2

12i 108

36. 8i 9 4i

46. The complex conjugate of 3

11 41i

5i.

1 5i 2

21 35i 6i 10i 2 21 41i 10

35. 12i 1 9i

5i 1

5i is 1

12 5 i 13 13

6 12i 7i 14i 2 1 4i 2 20 5i 4 i 5

NOT FOR SALE Section 2.4

57.

9 4i i Â&#x2DC; i i

9i 4i 2 i2

8 16i 2i 58. Â&#x2DC; 2i 2i 59.

4 5i

4 9i

16i 32i 2 4i 2

27i 120i 2 81 1600 120 27 i 1681 1681 60.

4i 2 9i 9 18i 16 4 9i 25 18i Â&#x2DC; 25 18i 25 18i

3i 9 40i Â&#x2DC; 9 40i 9 40i 120 27i 1681

100 72i 225i 162i 2 625 324 62 297i 62 297 i 949 949 949 64.

1 i 4 i

i 4 i

1 i 3 4 i i

5i 5 12i Â&#x2DC; 5 12i 5 12i

2 3 1 i 1 i

2 1 i 3 1 i

1 i 1 i

2 2i 3 3i 1 1 1 5i 2 1 5 i 2 2

62.

2i 5 2 i 2 i

5 7

5 Â&#x2DC;

66.

50i

i 2 i

6i 2i

12i 2

2 3 1

2 3

4i 2i 2 10 5i 4 i2 12 9i 5 12 9 i 5 5 69. 3

6 Â&#x2DC;

65.

2i 2 i 5 2 i

4 i 4i i 2 3i 4i i 2 5 1 4i Â&#x2DC; 1 4i 1 4i 5 20i 1 16i 2 5 20 i 17 17

25i 60i 2 25 144i 2 60 25i 60 25 i 169 169 169 61.

3 2i 3 8i

3i 8i 2 6i 4i 2 9 24i 6i 16i 2

5i 4 12i 9i 2

167

i 3 8i 2i 3 2i

i 2i 3 2i 3 8i

8 4i

3i 16 40i 25i 2

63.

Comp Complex Numbers

5i 7

10i

68.

15i

75i

5 2 1

15i 2

75i 2

5 2

21 3 10i 7 5i

21 50 7 21 5 2 7

67.

10i

50i 2

10 i

5 3 10 i 5 3

168

NOT FOR SALE

Chapter 2

70. 2

Polynomial ynomial and Rational Functions

6i 2

75. 4 x 2 16 x 15

4 2 6i 2 6i 6i 2

4 2 6i 2 6i 6 1

4 6 4 6i 2 4 6i 71. x 2 x 2 2

0; a

2 r 2r

1, b

2 1

4 1 2

0; a

1, b

6, c

6 4 1 10

77.

4, b

16, c

20 8

16, b

5 2

4, c

4 16 3

11 i 8

12 r

78.

2 9

6, c

4 9 37

1296 18 6 r 36i 1 r 2i 18 3

2 3

3, b 2

12, c

4 3 18

72 2r

7 2 3 5 x x 8 4 16

0 Multiply both sides by 16.

14 x 2 12 x 5

0; a

9, b

0; a

12 r 6 2i 6

0; a

0 Multiply both sides by 2.

16 4 4 17

2 4

74. 9 x 2 6 x 37

2 16

176 32

12 r

0; a

16 r 16 8 16 r 4i 8 1 2 r i 2

16 r 4 8

3 2 x 6x 9 2 3x 2 12 x 18 x

73. 4 x 2 16 x 17

6 r

4 r

1 r 8

2 6 2i 2 3 r i

12 8

4 r 4 11i 32

2 1

16 r

4 r

72. x 2 6 x 10

6 r

0; a

6 r

16 r 8

16, c

76. 16t 2 4t 3

2 2 r 2i 2 1ri

16 4 4 15

2 4

3 or x 2

4, b

16 r

2, c

0; a

12 r 12 r

2 14

14, b 2

12, c

4 14 5

136 28

12 r 2 34i 28 3 34 r i 7 14

NOT FOR SALE Section 2.4

79. 1.4 x 2 2 x 10

7 x 10 x 50 2

0; a

10 r 10 r

82. 4i 2 2i 3

0 Multiply both sides by 5.

10, c

7, b

10 4 7 50

83. 14i 5

14i 2i 2i

2 7

84. i

1 i 3

1500 85.

0; a

3, c

4.5, b

3 4 4.5 12

2 4.5

86.

87.

88.

6i 1 1 6i

1 i3

1 i 2i 1

89. 3i

90. i

9 16i, z2

1 1 z1 z2

Â§ 340 230i Â·Â§ 29 6i Â· Â¨ Â¸Â¨ Â¸ Â© 29 6i Â¹Â© 29 6i Â¹ 3

8i 2i 2i 2

1 i

1 i Â&#x2DC; i i

1 8i 3

1 8i 2i

81i 4

81i 2i 2

i 2i 2i 2

i i2

1 8i

1 8i Â&#x2DC; 8i 8i

8i 64i 2

81 1 1

1 1 1

1 i 8

20 10i

1 z

6i 2i i 2

8 1 1 1

23 i 3

8i 6

207

6 1 i 1

92. (a)

432 2i

3 r 3 23i 9

(b)

432 2 1 i

91. (a) z1

1 1 i

80. 4.5 x 3 x 12

3 r

1 i 2i

81. 6i 3 i 2

14i

5 5 15 r 7 7

1 r 3

14 1 1 i

4 2i

6 2i

6 2 i 216 2 2 i i

10 r 10 15 14

4 1 2 1 i

4i 2 2i 2i

169

Comp Complex Numbers

20 10i 9 16i 9 16i 20 10i

1 1 9 16i 20 10i

1 3

3 1

11,240 4630i 877 2

3i 3 1 3i

29 6i 340 230i

11,240 4630 i 877 877 2

1 3 3i 9i 2 3 3i 3 1 3 3i 9i 2 3 3i 2i 1 3 3i 9 3 3i 8 (b)

1 3

3 1 2

3i 3 1

1 3 3i 9i 2 3 3i 3 1 3 3i 9i 2 3 3i 2i 1 3 3i 9 3 3i 8

170

Chapter 2

Polynomial ynomial and Rational Functions

93. False.

94. True.

0 then a bi

If b

a bi

x 4 x 2 14

i 6 i 6

That is, if the complex number is real, the number equals its conjugate.

56 ?

36 6 14

95. False.

i 44 i150 i 74 i109 i 61

i 2

1 1 1 i i

i 2 i i 2 i 54

1 i 1 i 54

96. False.

98. (i) D

Sample answer: 4i 3 2i

3 6i, which is not a

(ii) F (iii) B

real number. i

97.

(iv) E

i 4i

(v) A (vi) C 99.

4 2

i 4i 3

i 4i 4

i 4i 4i

6i 6i

100. a1 b1i a2 b2i

6i 2

a1a2 a1b2i a2b1i b1b2i 2

a1a2

b1b2 a1b2 a2b1 i

The complex conjugate of this product is a1a2 b1b2 a1b2 a2b1 i.

4 4 2

iii

i11

i 4i 4i 3

i12

i 4i 4i 4

The product of the complex conjugates is

b1i a2 b2i

a1a2 a1b2i a2b1i b1b2i 2

a1a2

b1b2 a1b2 a2b1 i.

So, the complex conjugate of the product of two complex numbers is the product of their complex conjugates.

The pattern i, 1, i, 1 repeats. Divide the exponent by 4. If the remainder is 1, the result is i. If the remainder is 2, the result is 1. If the remainder is 3, the result is i. If the remainder is 0, the result is 1. 101. a1 b1i a2 b2i

a2 b1 b2 i

The complex conjugate of this sum is a1 a2 b1 b2 i. The sum of the complex conjugates is a1 b1i a2 b2i

a2 b1 b2 i.

So, the complex conjugate of the sum of two complex numbers is the sum of their complex conjugates.

NOT FOR SALE Section 2.5

Zeros of Polynomi Polynomial Polynom Functions

171

Section 2.5 Zeros of Polynomial Functions 18. f x

1. Fundamental Theorem of Algebra

4 x5 8 x 4 5 x3 10 x 2 x 2

2. Linear Factorization Theorem

Possible rational zeros: r1, r 2, r 12 , r 14

3. Rational Zero

Zeros shown on graph: 1, 12 , 12 , 1, 2

4. conjugate

19. f x

5. linear; quadratic; quadratic

x3 7 x 6

Possible rational zeros: r1, r 2, r 3, r 6

6. irreducible; reals

7. Descartesâ&#x20AC;&#x2122;s Rule of Signs

1 8. lower; upper

f x

9. Since f is a 1st degree polynomial function, there is one zero. 10. Since f is a 2nd degree polynomial function, there are two zeros. 11. Since f is a 3rd degree polynomial function, there are three zeros.

14. h t

1 t 1

20. f x

3 x 3 x 2

3 x 2 x 1

x3 13x 12

Possible rational zeros: r1, r 2, r 3, r 4, r 6, r12 3

f x

x x

3 x 3 x 4

3 x 4 x 1

x3 4 x 2 x 4 x 2 x 4 1 x 4

Since h is a 1st degree polynomial function, there is one real zero. x3 2 x 2 x 2

x x 22. h x

Zeros shown on graph: 2, 1, 1, 2

Possible rational zeros: r1, r 2, r 4, r 8, r16 Zeros shown on graph: 2, 2, 4

x3 9 x 2 20 x 12

1 1

h x

2 x 4 17 x3 35 x 2 9 x 45

Possible rational zeros: r1, r 3, r 5, r 9, r15, r 45, r 32 ,

4 x 1 x 1

Possible rational zeros: r1, r 2, r 3, r 4, r 6, r12

x3 4 x 2 4 x 16

r 12 ,

4 x 2 1

So, the rational zeros are 4, 1, and 1.

Possible rational zeros: r1, r 2

17. f x

So, the rational zeros are 2, 1, and 3.

21. g x

16. f x

So, the rational zeros are 3, 4, and 1.

t 2 2t 1 t 2 2t 1

15. f x

x x

12. Since f is a 7th degree polynomial function, there are seven zeros. 13. Since f is a 2nd degree polynomial function, there are two zeros.

0 3

r 52 ,

r 92 ,

r 15 , 2

r 45 2

x x

1 x 8 x 12

1 x 2 x 6

So, the rational zeros are 1, 2, and 6.

Zeros shown on graph: 1, 32 , 3, 5

172

NOT FOR SALE

Chapter 2

23. h t

Polynomial ynomial and Rational Functions

27. f x

t 3 8t 2 13t 6

Possible rational zeros: r1, r 2, r 3, r 6 6

t t

t 3 8t 2 13t 6

Ă2

6 t 1 t 1

1 p x

x x

f x

x x

3 x 2 6 x 9

28. f x

3 x 3 x 3

0 1

2 x3 3 x 2 1

x x

1 2 x 2 x 1

1 2 x 1

3 f x

Ă19

Ă9

Ă30

Ă10

x x

Ă12

Ă4

2 x 3 9 x 2 4

2 x 3 3 x 2 3 x 2

Ă25

Ă10

Ă5

Ă2

Ă5

Ă2

Ă3

Ă5

Ă3

Ă5

2 2

f x

Ă3

Ă5

Ă2

Ă5

5 x 1 x 1 2 x 5

29. z 4 z 3 z 2 3 z 6

Possible rational zeros: r1, r 2, r 3, r 6 1

3 3x 10 x 3

So, the rational zeros are 3 and 13 .

3 3 x 1 x 3

Ă4

So, the rational zeros are 5, 1, 1 and 52 .

3x3 19 x 2 33 x 9

Ă1

Possible rational zeros: r1, r 3, r 9, r 13 3

1 x 1 2 x 1

So, the rational zeros are 1 and 12 . 26. f x

Ă24

Ă15

Ă4

2 x 4 15 x3 23x 2 15 x 25

Possible rational zeros: r1, r 12 3

Ă27

Possible rational zeros: r1, r 5, r 25, r 12 , r 52 , r 25 2 5

So, the rational zeros are 2, 3, 23 , and 23 .

2 x3 3x 2 1

Ă18

So, the rational zero is 3. 25. C x

Ă58

Ă27

x3 9 x 2 27 x 27

Ă9

6 t 2 2t 1

Possible rational zeros: r1, r 3, r 9, r 27 3

Possible rational zeros: r1, r 2, r 3, r 4, r 6, r 8, r12, r 24, r 13 , r 23 , r 43 , r 83 , r 19 , r 92 , r 94 , r 89

So, the rational zeros are 1 and 6. 24. p x

9 x 4 9 x3 58 x 2 4 x 24

Ă6

1 z 3 2 z 2 3 z 6

1 z 2 3 z 2

So, the real zeros are Ă2 and 1.

NOT FOR SALE Section 2.5

30. x 4 13x 2 12 x

x x3 13 x 12

Zeros of Polynomi Polynomial Polynom Functions

33. f x

x3 x 2 4 x 4

(a) Possible rational zeros: r1, r 2, r 4

Possible rational zeros of x3 13 x 12:

(b) 4

r1, r 2, r 3, r 4, r 6, r12

Ã1

Ã12

Ã1

Ã12

Ã13

x x 1 x x 12

x x 1 x 4 x 3

â&#x2C6;&#x2019;6

r 12 ,

r1, r 2, r 4

Ã16

Ã4

Ã11

34. f x

3 x3 20 x 2 36 x 16

(a) Possible rational zeros: r1, r 2, r 4, r 8, r16, r 13 , r 23 , r 43 , r 83 , r 16 3 y

(b) 10 8 6 4 2

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019; 2

y y

â&#x20AC;&#x201C;4

Ã4

1 y 1 2 y 1 y 4

1 2

x x x 3 x 5 x 2

4 x3 15 x 2 8 x 3

(a) Possible rational zeros: r1, r 3, r 12 , r 32 , r 14 , r 34 0

and 1.

32. x x 3 x 5 x 2 x 4

(b)

Possible rational zeros of x 4 x3 3x 2 5 x 2:

r1, r 2

1 Ã2

8 10 12

â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;4

1 y 1 2 y 2 7 y 4

So, the real zeros are 4,

â&#x2C6;&#x2019;8

Possible rational zeros:

â&#x2C6;&#x2019;6

31. 2 y 3 y 16 y 15 y 4 3

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;4

The real zeros are 0, 1, 4, and 3. 4

173

Ã1

Ã3

Ã2

Ã3

Ã2

â&#x2C6;&#x2019;6 â&#x2C6;&#x2019; 4 â&#x2C6;&#x2019;2

x 2

8 10

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;6

x x 1 x 2 x 2 2 x 1

x x 1 x 2 x 1 x 1

The real zeros are 2, 0, and 1.

174

NOT FOR SALE

Chapter 2

36. f x

Polynomial ynomial and Rational Functions

40. f x

4 x3 12 x 2 x 15

(a) Possible rational zeros: r1, r 3, r 5, r15, r 12 , r 23 ,

4 x3 7 x 2 11x 18

(a) Possible rational zeros: r1, r 2, r 3, r 6, r 9, r18, 1 3 9 1 3 9 r ,r ,r ,r ,r ,r 2 2 2 4 4 4

r 52 , r 15 , r 14 , r 43 , r 54 , r 15 2 4 y

(b)

â&#x2C6;&#x2019;8

â&#x2C6;&#x2019; 24

â&#x2C6;&#x2019;9 â&#x2C6;&#x2019; 6 â&#x2C6;&#x2019;3

41. f x

5 2

2 x 4 13 x3 21x 2 2 x 8

(a) Possible rational zeros: r1, r 2, r 4, r 8, r 12

1 r 8

145 8

x 4 3x 2 2 r1, about r1.414

(a) x

(b) An exact zero is x

Ã3

Ã2

(b)

1 â&#x2C6;&#x2019;4

(c)

Ã1

â&#x2C6;&#x2019;8

Ã2

Ã1

Ã2

1 f x

4 x 4 17 x 2 4

(a) Possible rational zeros: r1, r 2, r 4, r 12 , r 14

1 x 1 x 2 2

1 x 1 x

2 x

(b)

42. P t

â&#x2C6;&#x2019;8

t 4 7t 2 12

r 2, about r1.732

(a) t â&#x2C6;&#x2019; 15

(b) An exact zero is t

Ã7

Ã6

Ã12

Ã3

Ã6

32 x3 52 x 2 17 x 3

(a) Possible rational zeros: r1, r 3, r 12 , r 23 , r 14 , r 43 , 1,r 3,r 1,r 3 r 18 , r 83 , r 16 16 32 32

Ã2

Ã3

Ã6

Ã2

Ã3

(b)

1 (c) P t

â&#x2C6;&#x2019;1

2 t 2 t 3

2 t 2 t

3 â&#x2C6;&#x2019;2

An exact zero is t

3 t

NOT FOR SALE Section 2.5

43. h x

x x 4 7 x3 10 x 2 14 x 24

(a) h x

(b) An exact zero is x

Note: f x

Ă7

Ă24

Ă12

Ă6

Ă4

Ă2

Ă4

Ă2

Ă8

Ă2

2 x

f x

Ă3

Ă27

Ă90

Ă30

6 (c) g x

49. If 3

x x

2 x 5 i

x 5 i

2 x 2 10 x 26

a x3 12 x 2 46 x 52 , where a is

48. If 3 2i is a zero, so is its conjugate, 3 2i.

f x

Ă30

Ă18

Ă9

Ă11

5 x 3 2i

x 3 2i

5 x 2 6 x 13

x3 11x 2 43x 65 Note: f x

a x3 11x 2 43 x 65 , where a is

any nonzero real number, has the zeros 5 and 3 r 2i.

3 x 3 6 x 2 11x 3

3 x 3 3x 1 2 x 3

2i is a zero, so is its conjugate, 3

3 x

2 x 1 ÂŞ x 3 ÂŹ

3 x

2 x 1 ÂŞÂŹ x 3

3x 2

2 x 2 ÂŞÂŤ x 3 ÂŹ

3x 2 3x 2

a x3 4 x 2 9 x 36 , where a is any

any nonzero real number, has the zeros 2 and 5 r i.

4 x 2 9

Note: f x

Ă51

An exact zero is x

4 x 3i x 3i

x3 12 x 2 46 x 52

Ă11

x x

47. If 5 i is a zero, so is its conjugate, 5 i.

r 3, 1.5, about 0.333

a x3 x 2 25 x 25 , where a is any

real number, has the zeros 4, 3i, and 3i.

6 x 4 11x3 51x 2 99 x 27

(b) An exact zero is x

f x

46. f x

Note: f x

(a) x

1 x 2 25

x3 4 x 2 9 x 36

x x 3 x 4 x

44. g x

1 x 5i x 5i

nonzero real number, has the zeros 1 and r 5i.

x x 3 x 4 x 2 2

h x

x x

175

x3 x 2 25 x 25

0, 3, 4, about r1.414

(c)

45. f x

x5 7 x 4 10 x3 14 x 2 24 x

Zeros of Polynomi Polynomial Polynom Functions

2i.

2i ÂşÂŞ x 3 ÂźÂŹ 2iÂşÂŞ ÂźÂŹ x 3

2i Âş Âź 2iÂşÂź

2i ÂşÂťÂź 2

x 2 x 2 6 x 9 2

x 2 x 2 6 x 11

3x 4 17 x3 25 x 2 23x 22 Note: f x

a 3 x 4 17 x3 25 x 2 23 x 22 , where a is any nonzero real number, has the zeros 23 , 1, and 3 r

2i.

176

Chapter 2

50. If 1

f x

NOT FOR SALE

Polynomial ynomial and Rational Functions

3i is a zero, so is its conjugate, 1

5 x 1 2

3i x 1

51. f x

3i.

10 x 25 x 2 2 x 4

x 8 x 9 x 10 x 100 4

Note: f x

(a) f x

9 x 2 3

(b) f x

9 x

a x 4 8 x3 9 x 2 10 x 100 , where

a is any real number, has the zeros 5, 5, and 1 r 52. f x

x 4 6 x 2 27

3 x

3i x 3i x

3 x

3i.

x 4 2 x3 3 x 2 12 x 18

x2 2x 3 x 6 x 2 x 3x 2 12 x 18 x4 6x2 2 x3 3x 2 12 x 2 x3 12 x 3x 2 18 3x 2 18 0 2

(a) f x

(b) f x

x x

6 x 2 2 x 3

6 x

6 x 1

6 x 2 2 x 3

2i x 1

Note: Use the Quadratic Formula for (c). 53. f x

x 4 4 x3 5 x 2 2 x 6

x 2x 2 x 4x x 4 2 x3 2 x3 2 x3 2

(a) f x

(b) f x

x 1 x 1

x2 5x2 2x2 7 x2 4x2 3x 2 3x 2

2x 3 2x 6

2x 4x 6x 6 6x 6 0

2 x 2 x 2 2 x 3

3 x 1

3 x 2 2 x 3

2i x 1

Note: Use the Quadratic Formula for (b) and (c).

NOT FOR SALE Section 2.5

54. f x

Zeros of Polynomi Polynomial Polynom Functions

177

x 4 3 x3 x 2 12 x 20 x 2 3x 5

x 4 x 3x x 2 12 x 20 4 x2 x4 3 3x 5 x 2 12 x 3x3 12 x 5 x 2 20 5 x 2 20 0 2

(a) f x

4 x 2 3 x 5

(b) f x

Note: Use the Quadratic Formula for (b). 55. f x

x3 x 2 4 x 4

Alternate Solution:

Ã1

Ã4

4 2i

2i 1

1 f x

2i 1

Ã2i

Ã1

56. f x

2 2

f x

x 4x 4x 0x 0x

1 4 4 4 0

4 x 1

The zeros of f x are x

1, r 2i.

2 x3 3 x 2 18 x 27

2 Ã3i

x 2 4 is a factor of f x .

x 2 0 x 4 x3 x 2 x3 0 x 2 x2 x2

Because 3i is a zero, so is 3i. 3i

2i x 2i

By long division, you have:

2i x 2i x 1

The zeros of f x are x

r 2i are zeros of f x ,

Because x

Because 2i is a zero, so is Ã2i.

Alternate Solution:

9i 18

Ã27

3 6i

Because x

r 3i are zeros of f x , x 2 9 is a factor of f x .

3i x 3i

By long division, you have: 2x 3

3 6i

Ã9i

x 0 x 9 2 x 3x 2 x3 0 x 2 3x2 3x2 2

3i x 3i 2 x 3

The zeros of f x are x

r 3i, 32 .

f x

18 x 27 18 x 0 x 27 0 x 27 0

9 2 x 3

The zeros of f x are x

r 3i, 32 .

178

NOT FOR SALE

Chapter 2

57. f x

Polynomial ynomial and Rational Functions

2 x 4 x3 49 x 2 25 x 25

Because 5i is a zero, so is Ă5i. 5i

Ă5i

Ă1

Ă25

10i

5i 50

5i 25

1 10i

1 5i

Ă5i

1 10i

1 5i

Ă5i

10i

Ă1

2 f x

x x

5i x 5i 2 x x 1

5i x 5i 2 x 1 x 1

The zeros of f x are x

r5i, 12 , 1.

Alternate Solution: r 5i are zeros of f x , x 5i x 5i

Because x

x 2 25 is a factor of f x .

By long division, you have: 2 x2 x 0 x 25 2 x x 2 x 4 0 x3 x3 x3 2

f x

25 2 x 2 x 1

The zeros of f x are x 58. g x

x 1

49 x 2 25 x 25 50 x 2 x 2 25 x 0 x 2 25 x x 2 0 x 25 x 2 0 x 25 0

r 5i, 12 , 1.

x3 7 x 2 x 87

Because 5 2i is a zero, so is 5 2i. 5 2i

5 2i

Ă1

5 2i

14 6i

Ă87

2 2i

15 6i

2 2i

15 6i

5 2i

15 6i

Ă7

1 The zero of x 3 is x

3. The zeros of f x are x

3, 5 r 2i.

NOT FOR SALE Section 2.5

59. g x

4 x3 23 x 2 34 x 10

Because 3 r i are zeros of g x ,

3 i

Ă10

12 4i

37 i

11 4i

3 i

11 4i

3 i

12 4i

3 i

Ă1

4 The zero of 4 x 1 is x g x are x

ÂŞÂŹ x 3 i ÂşÂŞ ÂźÂŹ x 3 i ÂşÂź

4x 1 x 2 6 x 10 4 x3 23 x 2 4 x3 24 x 2 x2 x2

. The zeros of

3 r i, 14 .

34 x 10 40 x 6 x 10 6 x 10 0

6 x 10 4 x 1

3 r i, 14 .

3x3 4 x 2 8 x 8

Because 1

is a factor of g x . By long division, you have:

The zeros of g x are x

3 i 2

x 2 6 x 10

g x

60. h x

ÂŞÂŹ x 3 iÂşÂŞ ÂźÂŹ x 3 iÂşÂź

0 1 4

179

Alternate Solution

Because 3 i is a zero, so is 3 i. 3 i

Zeros of Polynomi Polynomial Polynom Functions

3i is a zero, so is 1

3i.

Ă4

3 3 3i

10 2 3i

Ă8

1 3 3i

2 2 3i

1 3 3i

2 2 3i

3 3 3i

2 2 3i

3 The zero of 3x 2 is x

23 . The zeros of f x are x

23 , 1 r

3i.

180

NOT FOR SALE

Chapter 2

61. f x

Polynomial ynomial and Rational Functions

x 4 3 x3 5 x 2 21x 22 2i is a zero, so is 3

Because 3

ÂŞ x 3 ÂŹ

2i ÂşÂŞ x 3 ÂźÂŹ

2i, and ÂŞ x 3 ÂŹ

2i Âş Âź

3 2

2iÂşÂŞ ÂźÂŹ x 3

2iÂşÂź

x 2 6 x 11 is a factor of f x . By long division, you have: x 2 3x 2 x 6 x 11 x 3 x 5 x 2 21x 22 2

x 4 6 x3 11x 2 3 x3 16 x 2 21x 3 x3 18 x 2 33x 2 x 2 12 x 22 2 x 2 12 x 22 0

x2 x2

f x

6 x 11 x 2 3 x 2

6 x 11 x 1 x 2

The zeros of f x are x 62. f x

3 r

2i, 1, 2. 65. h x

x3 4 x 2 14 x 20

By the Quadratic Formula, the zeros of f x are

Because 1 3i is zero, so is 1 3i. 1 3i

1 3i

12 6i

Ă20

3 3i

2 6i

3 3i

2 6i

1 3i

2 6i

1 The zero of x 2 is x

The zeros of f x are x

2, 1 r 3i.

63. f x

x 2 36

64. f x

x f x

66. g x

223i 2

4 68 2

64 2

1 r 4i.

x 1 4i

1 4i x 1 4i

x 2 10 x 17

By the Quadratic Formula, the zeros of f x are: x

67. f x

10 r

100 68 2

10 r 2

5 r 2

x 5 2 2

x 5 2 2 x 5 2 2

By the Quadratic Formula, the zeros of f x are 1 224 2

f x

r 6i.

x 2 x 56

f x

6i x 6i

The zeros of f x are x

x 2 2 x 17

x 4 16

x2 x

4 x 2 4

2 x 2 x 2i x 2i

Zeros: r 2, r 2i

Â§ 1 223i ÂˇÂ§ 1 223i Âˇ Â¸Â¨ x Â¨Â¨ x Â¸Â¨ Â¸Â¸ 2 2 ÂŠ ÂšÂŠ Âš

NOT FOR SALE Section 2.5

68.

f y

72. f x

y 4 256

y2 y

Zeros of Polynomi Polynomial Polynom Functions

16 y 2 16

x3 x 2 x 39

Possible rational zeros: r1, r 3, r13, r 39

4 y 4 y 4i y 4i

Zeros: r 4, r 4i 1

69. f z

z2 2z 2

By the Quadratic Formula, the zeros of f z are z

f z

4 8 2

1 r i.

70. h x

1 i z 1 i

Ă3

Ă2

are x

4 8 2

71. g x

1 r i.

h x

16 52 2

2 r 3i

3 x 2 3i x 2 3i

4 12 2

2i.

2 ÂŞ x 1 ÂŹ

2 x 1

2i x 1

2i ÂşÂŞ x 1 ÂźÂŹ 2i

2i Âş Âź

x3 9 x 2 27 x 35

Possible rational zeros: r1, r 5, r 7, r 35 5

2ri

1 1

1 x 2 i x 2 i

By the Quadratic Formula, the zeros of x 2 4 x 7 are x

Zeros: 1, 2 r i g x

16 20 2

x3 x 6

74. h x

By the Quadratic Formula, the zeros of x 2 4 x 5 are: x

Zeros: 2, 1 r

1 x 1 i x 1 i

By the Quadratic Formula, the zeros of x 2 2 x 3 are

x3 3x 2 x 5

Possible rational zeros: r1, r 5 1

f x

Zeros: 1, 1 r i h x

Possible rational zeros: r1, r 2, r 3, r 6

By the Quadratic Formula, the zeros of x 2 2 x 2 2r

73. h x

x3 3x 2 4 x 2

Zeros: 3, 2 r 3i

Possible rational zeros: r1, r 2 1

By the Quadratic Formula, the zeros of x 2 4 x 13 are: x

ÂŞÂŹ z 1 i ÂşÂŞ ÂźÂŹ z 1 i ÂşÂź

181

4 r

16 28 2

Zeros: 5, 2 r h x

2 r

3i.

5 x 2

3i x 2

182

NOT FOR SALE

Chapter 2

75. f x

Polynomial ynomial and Rational Functions

78. h x

5 x3 9 x 2 28 x 6

Possible rational zeros: r1, r 3, r 9

Possible rational zeros: 1 2 3 6 r1, r 2, r 3, r 6, r , r , r , r 5 5 5 5 15

5 5

4 24 2

1 Zeros: , 1 r 5

h x

5i ÂşÂŞ x 1 ÂźÂŹ

5i x 1

5i Âş Âź

16 112 4

9 0

0 r i.

3 x i x i

x 4 10 x 2 9

1 x 2 9

i x i x 3i x 3i

Zeros: r i, r 3i 80. f x

x 4 29 x 2 100

x2 x

25 x 2 4

2i x 2i x 5i x 5i

Zeros: r 2i, r 5i

96 4

81. f x

x3 24 x 2 214 x 740

Possible rational zeros: r1, r 2, r 4, r 5, r10, r 20, r 37, r 74, r148, r185, r 370, r 740

6i. 2000

3 Zeros: , 1 r 2

3Âˇ Â§ Â¨x Â¸ x 1 2Âš ÂŠ

f x

By the Quadratic Formula, the zeros of 2 x 2 4 x 14 are x

Possible rational roots: 1 3 7 21 r , r1, r , r 3, r , r 7, r , r 21 2 2 2 2 1

79. f x

2 x3 x 2 8 x 21

Zeros: 3, r i

5 x 1 x 1

3 2

The zeros of x 2 1 are x

76. g x

1 1

5i.

ÂŞ Â§ 1 ÂˇÂş ÂŞ ÂŤ x Â¨ 5 Â¸Âť 5 ÂŹ x 1 ÂŠ ÂšÂź ÂŹ

f x

1 1

By the Quadratic Formula, the zeros of 5 x 2 10 x 30 5 x 2 2 x 6 are x

x 4 6 x3 10 x 2 6 x 9

6i x 1

â&#x2C6;&#x2019; 20

â&#x2C6;&#x2019;1000

77. g x

x 4 x 8 x 16 x 16 4

Possible rational zeros: r1, r 2, r 4, r 8, r16 2

2 1

1 g x

16 0

2 x 2 x 2 4

2 x 2i x 2i

Based on the graph, try x 10

1 1

10.

214

740

140

740

By the Quadratic Formula, the zeros of x 2 14 x 74 are x

14 r

196 296 2

The zeros of f x are x

7 r 5i.

10 and x

7 r 5i.

INSTRUCTOR USE ONLY Zeros: 2, r 2i Zeros 2i

NOT FOR SALE Section 2.5

82. f s

Zeros of Polynomi Polynomial Polynom Functions

84. f x

2 s 3 5s 2 12 s 5

1 5 Possible rational zeros: r1, r 5, r , r 2 2

9 x3 15 x 2 11x 5

1 5 1 5 Possible rational zeros: r1, r 5, r , r , r , r 3 3 9 9 5

â&#x2C6;&#x2019;10

â&#x2C6;&#x2019;5

â&#x2C6;&#x2019;10

Based on the graph, try s 1 2

2 2

Based on the graph, try x

1 . 2

4 20 2

83. f x

By the Quadratic Formula, the zeros of 9 x 2 6 x 5

1 r 2i.

are x

36 180 18

1 2 r i. 3 3

The zeros of f x are x

1 and s 2

The zeros of f s are s

9 9

By the Quadratic Formula, the zeros of 2 s 2 2 s 5

are s

1 r 2i.

85. f x

16 x3 20 x 2 4 x 15

2 x 4 5 x3 4 x 2 5 x 2

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;5

2 and x

Based on the graph, try x â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;5

Based on the graph, try x 16

3 . 4

64 80 8

2 2

1 2

2 0

The zeros of 2 x 2 2

By the Quadratic Formula, the zeros of 16 x 2 32 x 20 4 4 x 2 8 x 5 are 8r

1 2

3 4

1 2 r i. 3 3

1 and x

Possible rational zeros: r1, r 2, r

Possible rational zeros: 1 3 5 15 1 3 r1, r 3, r 5, r15, r , r , r , r , r , r , 2 2 2 2 4 4 5 15 1 3 5 15 1 3 5 15 r ,r ,r ,r ,r ,r ,r ,r ,r ,r 4 4 8 8 8 8 16 16 16 16

183

The zeros of f x are x

1 1 r i. 2

The zeros of f x are x

3 and x 4

2 x 2 1 are x 2, x

1 . 2

r i.

1 , and x 2

r i.

1 i. 2

184

Chapter 2

86. g x

NOT FOR SALE

Polynomial ynomial and Rational Functions

x5 8 x 4 28 x3 56 x 2 64 x 32

Possible rational zeros: r1, r 2, r 4, r 8, r16, r 32

91. g x

5 x x 4 2

5 x5 10 x

Let g x

x 4 2.

Sign variations: 1, positive zeros: 1 â&#x2C6;&#x2019;10

g x

x4 2

Sign variations: 1, negative zeros: 1 â&#x2C6;&#x2019;10

92. f x

Based on the graph, try x 2

Sign variations: 3, positive zeros: 3 or 1 f x

93. f x

4 16 2

The zeros of g x are x 87. g x

5 x3 x 2 x 5

Sign variations: 3, positive zeros: 3 or 1 f x

94. f x

3x3 2 x 2 x 3

Sign variations: 0, positive zeros: 0

3i.

95. f x

(a)

x3 3x 2 2 x 1

2 x3 3x 2 3

4x2 8x 3

96. f x

x3 4 x 2 1

1 1

2 x3 3 x 2 1

4 is an upper bound. (b)

2 x 4 3x 2

Sign variations: 2, positive zeros: 2 or 0 h x

4 is a lower bound.

(a)

2 x3 3 x 2 1

Sign variations: 1, negative zeros: 1 90. h x

Sign variations: 0, positive zeros: 0 h x

1 1

4x2 8x 3

Sign variations: 0, negative zeros: 0 89. h x

(b)

Sign variations: 2, positive zeros: 2 or 0 h x

1 is an upper bound.

Sign variations: 0, negative zeros: 0 88. h x

3 x3 2 x 2 x 3

Sign variations: 3, negative zeros: 3 or 1

3i.

2 and x

f x

Sign variations: 1, positive zeros: 1 g x

5 x3 x 2 x 5

Sign variations: 0, negative zeros: 0

By the Quadratic Formula, the zeros of x 2 x 4 2r

4 x3 3 x 2 2 x 1

Sign variations: 0, negative zeros: 0

are x

4 x3 3x 2 2 x 1

2 x 4 3x 2

1 is a lower bound.

Sign variations: 0, negative zeros: 0

NOT FOR SALE Section 2.5

97. f x

(a)

101. f y

x 4 4 x3 16 x 16

205

189

1 3

1 1

Possible rational zeros: r1, r 2, r 3, r 6, r 12 , r 32 , r 14 , r 34 34

141

125

y 34 4 y y 34 4 y

4 y3 3 y 2 8 y 6

3 is a lower bound. 98. f x

(a)

4 y

2 x4 8x 3

2 2

138

141

102. g x

128

544

136

547

3 is a lower bound. 99. f x

4 x3 3x 1

4 4

4 x3 3x 1

2 3

f z

1 4 x 2 4 x 1

1 2 x 1

3 y 2 2

x 23 3x

3 x

2 x 2 5

103. P x

1 4

104. f x

2 x

25 2 x 4

25 x 2 36

9 x 2 4

3 2 x 3 x 2 x 2

The rational zeros are r 32 and r 2.

1 2

2 x3 3x 2

23x 12

Possible rational zeros: r1, r 2, r 3, r 4, r 6, r12, r 12 , 4

f x

1 2

4 2 x 2 5 x 3

1 2

4 2 x 1 x 3

3 2

The rational zeros are 3, 12 , and 4.

2 z 32 6 z 2 7 z 3

2 z

2 3

1 r 14 , r 34 , r 94 , r 16 , r 12

4 x 1 4 2 x 4

12 z 3 4 z 2 27 z 9

1 4

Possible rational zeros: r1, r 3, r 9, r 12 , r 32 , r 92 , r 13 ,

3 2

So, the only real zero is 23 .

1 So, the zeros are 1 and . 2 100. f z

3 x3 2 x 2 15 x 10

g x

Possible rational zeros: r1, r 12 , r 14 1

Possible rational zeros: r1, r 2, r 5, r10, r 13 , r 23 , r 53 , r 10 3

So, the only real zero is 34 .

3 is an upper bound. (b)

185

4 y3 3 y 2 8 y 6

5 is an upper bound. (b)

Zeros of Polynomi Polynomial Polynom Functions

3 3 z 1 2 z 3

So, the real zeros are 32 , 13 , and 32 .

186

NOT FOR SALE

Chapter 2

105. f x

Polynomial ynomial and Rational Functions

x3 1 4

1 x2 4

4 x3 x 2 4 x

1 ÂŞx 2 4ÂŹ

4 x 1

1 4

4 x

1 x 2 1

1 4

4 x

1 x 1 x 1

1 4

1 6

6 z

Matches (d). 108. f x

x3 2

and r1.

2 x2

Rational zeros: 0

11z 3z 2

Irrational zeros: 1 x Matches (a). 109. f x

x x 1 x 1

Rational zeros: 3 x

Irrational zeros: 0

6 f x

1 x 2 x 1

Irrational zeros: 0

Possible rational zeros: r1, r 2, r 12 , r 13 , r 23 , r 16 2

x3 1

Rational zeros: 1 x

1 1 4 x 1 ÂşÂź

The rational zeros are 106. f z

107. f x

1 4

1 6

2 6 x x 1

1 6

2 3 x 1 2 x 1

x3 x

0, r1

Matches (b).

110. f x

x3 2 x x x 2 2

The rational zeros are 2, 13 , and 12 .

x x

2 x

Rational zeros: 1 x

Irrational zeros: 2 x

Matches (c). 15

111. (a)

(b) V 9â&#x2C6;&#x2019;

x â&#x2C6;&#x2019;2

V 125

Volume of box

2 x 9 2 x x

Because length, width, and height must be positive, you have 0 x 92 for the domain.

(c)

l Â&#x2DC;wÂ&#x2DC;h

x 9 2 x 15 2 x

100

(d) 56

x 9 2 x 15 2 x

135 x 48 x 2 4 x3

4 x3 48 x 2 135 x 56

The zeros of this polynomial are 12 , 72 , and 8.

50 25

x cannot equal 8 because it is not in the domain of V.

x 1

[The length cannot equal 1 and the width cannot equal 7. The product of 8 1 7 56 so it

Length of sides of squares removed

The volume is maximum when x | 1.82. The dimensions are: length | 15 2 1.82

width | 9 2 1.82

height

showed up as an extraneous solution.] 11.36 5.36

So, the volume is 56 cubic centimeters when x centimeter or x

x | 1.82

7 2

1 2

centimeters.

1.82 cm u 5.36 cm u 11.36 cm

NOT FOR SALE Section 2.5 117. g x

112. (a) Combined length and width:

4x y

120 Â&#x; y

Volume

lÂ&#x2DC;wÂ&#x2DC;h

120 4 x

187

f x . This function would have the same

zeros as f x , so r1 , r2 , and r3 are also zeros of g x .

x2 y

118. g x

x 2 120 4 x

3 f x . This function has the same zeros as f

because it is a vertical stretch of f. The zeros of g are r1 , r2 , and r3.

4 x 2 30 x

(b)

Zeros of Polynomi Polynomial Polynom Functions

18,000

119. g x

f x 5 . The graph of g x is a horizontal

shift of the graph of f x five units of the right, so the 0

zeros of g x are 5 r1 , 5 r2 , and 5 r3 .

30 0

120. g x

Dimensions with maximum volume: 20 in. u 20 in. u 40 in.

x3 30 x 2 3375

1 1

4 x 30 x

4 x 120 x 13,500 3

121. g x

3375

225

3375

225

3 f x . Because g x is a vertical shift of

determined. 122. g x

f x . Note that x is a zero of g if and only if

x is a zero of f. The zeros of g are r1 , r2 , and r3 .

Using the Quadratic Formula, x The value of

r1 r2 r , , and 3 . 2 2 2

the graph of f x , the zeros of g x cannot be

15 x 2 15 x 225

2x is a zero of f. The zeros of g are

13,500

(c)

f 2 x . Note that x is a zero of g if and only if

15 r 15 5 15, . 2

123. Zeros: 2, 12 , 3 f x

x 2 2 x 1 x 3

2 x 3 3x 2 11x 6

15 15 5 is not possible because it is 2

negative. 113. (a) Current bin: V V New bin:

V x

2u3u4 24 cubic feet 5 24 120 cubic feet

2 x 3 x 4

(b) x3 9 x 2 26 x 24 x3 9 x 2 26 x 96

(â&#x2C6;&#x2019; 2, 0) â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;8

120

(3, 0) x 4

120 0 Any nonzero scalar multiple of f would have the same af x , a ! 0. There are three zeros. Let g x

The only real zero of this polynomial is x 2. All the dimensions should be increased by 2 feet, so the new bin will have dimensions of 4 feet by 5 feet by 6 feet.

infinitely many possible functions for f. 124.

114. C

( 12 , 0(

x Âˇ Â§ 200 100Â¨ 2 Â¸, x t 1 x 30 Âš ÂŠ x

C is minimum when 3x3 40 x 2 2400 x 36000

y 50

(â&#x2C6;&#x2019;1, 0)

The only real zero is x | 40 or 4000 units.

(1, 0)

(4, 0) x

(3, 0) 4

115. False. The most complex zeros it can have is two, and the Linear Factorization Theorem guarantees that there are three linear factors, so one zero must be real. 116. False. f does not have real coefficients.

188

Chapter 2

NOT FOR SALE

Polynomial ynomial and Rational Functions

Section 2.6 Rational Functions 7. Because the denominator is zero when x 2 1 0, 1 and the domain of f is all real numbers except x

1. rational functions 2. vertical asymptote

3. horizontal asymptote

x 4. slant asymptote

f x

5. Because the denominator is zero when x 1 domain of f is all real numbers except x 1.

x f x

0.5

0.9

0.99

100

o f

1.01

1.1

1.5

f m

100

o 1

5.4

17.3

152.3

1 m

0.99

0.9

0.5

f m

147.8

12.8

As x approaches 1 from the left, f x increases

x f x

6. Because the denominator is zero when x 2 0, the 2. domain of f is all real numbers except x

f x

1.01

f x decreases without bound.

without bound.

1.1

without bound. As x approaches 1 from the right,

bound. As x approaches 1 from the right, f x increases

f x

1.5

0, the

As x approaches 1 from the left, f x decreases without

f x

0.5

0.9

0.99

12.8

147.8

o f

1.01

1.1

1.5

f m

152.3

17.3

5.4

As x approaches 1 from the left, f x increases without

2.5

2.1

2.01

o 2

bound. As x approaches 1 from the right, f x

105

1005

o f

decreases without bound.

2 m

1.99

1.9

1.5

f m

955

As x approaches 2 from the left, f x increases without bound. As x approaches 2 from the right, f x increases without bound.

8. Because the denominator is zero when x 2 4 0, the domain of f is all real numbers except x 2 and x 2.

x f x

2.5

2.1

2.01

1.2

2.2

10.2

100.2

1.99

1.9

1.5

Undef.

99.7

9.7

1.7

0.7

As x approaches 2 from the left, f x decreases without bound. As x approaches 2 from the right, f x increases without bound. x f x

x f x

1.5

1.9

1.99

0.7

1.7

9.7

99.7

2.01

2.1

2.5

Undef.

100.2

10.2

2.2

1.2

As x approaches 2 from the left, f x decreases without bound. As x approaches 2 from the right, f x increases without bound.

Section 2.6 9. f x

4 x2

Vertical asymptote: x

Vertical asymptote: None

Horizontal asymptote: y

ÂŞÂŹDegree of N x degree of D x ÂşÂź

10. f x

ÂŞÂŹDegree of N x

Domain: All real numbers. The denominator has no real zeros. [Try the Quadratic Formula on the denominator.]

Horizontal asymptote: y

Vertical asymptote: None

ÂŞÂŹDegree of N x degree of D x ÂşÂź

5 x 5 x

Vertical asymptote: x

3 7x 3 2x

Horizontal asymptote: y

3 2

7 x 3 2x 3

Horizontal asymptote: y

Â§ 1Âˇ (b) y-intercept: Â¨ 0, Â¸ ÂŠ 2Âš

degree of D x ÂşÂź

Vertical asymptote: x

degree of D x ÂşÂź

1 x 2

17. f x

(a) Domain: all real numbers x except x

Domain: all real numbers except x

ÂŞÂŹDegree of N x

Horizontal asymptote: y ÂŞÂŹDegree of N x

Horizontal asymptote: y

x 5 x 5

Domain: all real numbers except x

13. f x

degree of D x ÂşÂź

3x 2 x 5 x2 1

16. f x

Vertical asymptote: x

12. f x

Domain: all real numbers except x

11. f x

Domain: All real numbers. The denominator has no real zeros. [Try the Quadratic Formula on the denominator.]

Horizontal asymptote: y

189

3x 2 1 x x 9

15. f x

Domain: all real numbers except x

Ration Rational Functions

(d)

f x

1 2

1 3

7 2

degree of D x ÂşÂź

â&#x2C6;&#x2019;3

x3 2 x 1

(0, 12 ( x

â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1

Domain: all real numbers except x Vertical asymptotes: x

â&#x2C6;&#x2019;2

Horizontal asymptote: None ÂŞÂŹDegree of N x ! degree of D x ÂşÂź

14. f x

4 x2 x 2

Vertical asymptote: x

Horizontal asymptote: None ÂŞÂŹDegree of N x

degree of D x ÂşÂź

190

NOT FOR SALE

Chapter 2

Polynomial ynomial and Rational Functions

1 x 3

18. f x

(a) Domain: all real numbers x except x

f x

1 3

1 2

1 3

(d)

x g x

x 2

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;1

(0, â&#x2C6;&#x2019; 13(

1 8

1 6

1 4

1 2

)0, 16 ) x 2

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;6

1 x 4

7 2x 2 x

21. C x

(a) Domain: all real numbers x except x

h x

1 2

(a) Domain: all real numbers x except x

(d)

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;3

19. h x

â&#x2C6;&#x2019;2

Horizontal asymptote: y

1Â· Â¸ 6Â¹

Horizontal asymptote: y x

1 x 6

(a) Domain: all real numbers x except x

(d)

1 6 x

20. g x

0 2

1 2

1 3

1 4

Horizontal asymptote: y (d)

C x

1 2

7 2

4 3

2 1 â&#x2C6;&#x2019; 7 â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;5

â&#x2C6;&#x2019;3

6 x

â&#x2C6;&#x2019;1

)0, â&#x2C6;&#x2019; 14 )

)0, 72 )

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;4

1 â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;5 â&#x2C6;&#x2019;4 â&#x2C6;&#x2019; 7, 0 2

)

â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;2

NOT FOR SALE Section 2.6

1 3x 1 x

22. P x

3x 1 x 1

(d)

Horizontal asymptote: y 1

P x

(a) Domain: all real numbers s

y-intercept: 0, 1

(b) Intercept: 0, 0

(d)

191

4s s 4

24. g s

(a) Domain: all real numbers x except x

Ration Rational Functions

g s

4 5

y 4

3 2

â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2

(0, 0) 2

â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;4

( 13 , 0)

(0, 1) â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;1

1 2t t

25. f t

x 4

2t 1 t

(a) Domain: all real numbers t except t x2 x2 9

23. f x

(a) Domain: all real numbers x

(b) Intercept: 0, 0

0 2

Horizontal asymptote: y

(d)

f x

1 10

4 13

1 2

f t

1 2

5 2

y y

â&#x2C6;&#x2019;2

3 2

( 12 , 0) t

â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;1 2

â&#x2C6;&#x2019;3

(0, 0) â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;1

192

Chapter 2

26. f x

NOT FOR SALE

Polynomial ynomial and Rational Functions 1

(a) Domain: all real numbers x except x

Horizontal asymptote: y

Horizontal asymptote: y 1 2 4 9

f x

1 4

8Â· Â¸ 9Â¹

(d) 0

(b) x-intercepts: 4, 0 , 2, 0

(a) Domain: all real numbers x except x

3 2

5 2

7 2 4 9

(d)

1 4

g x

27 16

16 7

8 9

8 5

7 16

(

0, â&#x2C6;&#x2019; 14

(

â&#x2C6;&#x2019;1

(0, 89 (

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;4

(â&#x2C6;&#x2019;2, 0) â&#x2C6;&#x2019;3

(4, 0) x 2

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;4

x 1 x 4

x 2 x 2

x2 5x 4 x2 4

27. h x

29. f x

(b) x-intercepts: 1, 0 , 4, 0

2, x

Horizontal asymptote: y

h x

10 3

28 5

10 3

y 6

1 ,x z 4 x 4 r4

Because x 4 is a D x , x 4 is not a

vertical asymptote of f x .

Horizontal asymptote: y

Â¬ÂªDegree of N x degree of D x ÂºÂ¼

0 1

2 5

4 0

30. f x

x 1 x2 1

x 1 1 x 1

1 , x z 1 x 1

Domain: all real numbers x except x r1 Vertical asymptote: x 1 (Because x 1 is a common factor of N x and D x , x 1 is not a vertical asymptote of f x . )

â&#x2C6;&#x2019;4

x 4 4 x 4

common factor of N x and

â&#x2C6;&#x2019;6

Vertical asymptote: x

y-intercept: 0, 1

x 4 x 2 16

Domain: all real numbers x except x

(a) Domain: all real numbers x except x

(d)

x 4 x x 3 x

x2 2 x 8 x2 9

28. g x

Horizontal asymptote: y

(1, 0) x

(4, 0) 6

ÂªÂ¬Degree of N x degree of D x ÂºÂ¼

(0, â&#x2C6;&#x2019;1)

NOT FOR SALE Section 2.6

t2 1 t 1

31. f t

1 t 1

t 1,

t 1

(b) t-intercept: 1, 0

4 3

y-intercept: 0, 1

(â&#x2C6;&#x2019;1, 0)

â&#x2C6;&#x2019; 4 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2

f t

Undef.

x 2 36 x 6

32. f x

6 x 6

(0, 1) t 1

â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;4

x 6, x z 6

x 6

(a) Domain: all real numbers x except x

(b) x-intercept: 6, 0

4 2

y-intercept: 0, 6

f x

â&#x2C6;&#x2019;6

Undef.

x 5 x 5

x 5 x 1

x 2 25 x 4x 5

33. f x

Domain: all real numbers x except x

x 5 ,x z 5 x 1

5 and x

common factor of N x and D x , x

Horizontal asymptote: y

(0, â&#x2C6;&#x2019;6)

x2 4 x 3x 2 x 2 x 2

2 x 1

asymptote of f x . )

degree of D x ÂºÂ¼

x 2 3x x x 6

Horizontal asymptote: y

x x 3

3 x 2

(a) Domain: all real numbers x except x

x , x 2 3 and x

1 3

0 0

Horizontal asymptote: y

f x

degree of D x ÂºÂ¼

x z 3

(b) Intercept: 0, 0

x 2 ,x z 2 x 1

Domain: all real numbers x except x 1 and x 2 Vertical asymptote: x 1 (Because x 2 is a common factor of N x and D x , x 2 is not a vertical

5 is not a

ÂªÂ¬Degree of N x

(d)

8 10

â&#x2C6;&#x2019; 12

34. f x

vertical asymptote of f x .

35. f x

â&#x2C6;&#x2019; 10

1 Because x 5 is a

Vertical asymptote: x

ÂªÂ¬Degree of N x

x 2

â&#x2C6;&#x2019;4

(6, 0)

â&#x2C6;&#x2019;6 â&#x2C6;&#x2019; 4 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;2

193

t z 1

(a) Domain: all real numbers t except t

(d)

Ration Rational Functions

1 1

3 3

4 2

â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;2

(0, 0) â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;6

194

Chapter 2

NOT FOR SALE

Polynomial ynomial and Rational Functions

5 x 4

4 x 3

5 , x 3

(a) Domain: all real numbers x except x

4 or x

36. f x

x x 12 2

x z 4 3

y 6

Horizontal asymptote: y

4 2

(d)

x f x

37. f x

5 3

x 2 3x 4 2 x2 x 1 x 1 x 4

2 x

5 2

5 4

(

1 and x 2

1 (Because x 1 is a 2 common factor of N x and D x , x 1 is not a

38. f x

(d)

1 2

2 x

1 x 3

2 x 1 x 1

2, x

Horizontal asymptote: y

degree of D x ÂºÂ¼

6 x 2 11x 3 6 x2 7 x 3 2 x 3 3x 1

2 x

Â§ 1 Â· (b) x-intercepts: Â¨ , 0 Â¸, 3, 0

Â© 2 Â¹

vertical asymptote of f x . )

ÂªÂ¬Degree of N x

2 x2 5x 3 x 2 x2 x 2 3

(a) Domain: all real numbers x except x

Vertical asymptote: x

Horizontal asymptote: y

â&#x2C6;&#x2019;6

x 4 , x z 1 2x 1

Domain: all real numbers x except x

(

39. f x

1 x 1

0, â&#x2C6;&#x2019; 5 3 â&#x2C6;&#x2019;4

f x

3 4

5 4

1, and x

3 2

3 2 48 5

3 10

3x 1 3 ,x z 3x 1 2

3 3x 1

Domain: all real numbers x except 3 1 x or x 2 3

(

â&#x2C6;&#x2019; 1, 0 2

(

6 3

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;3

(3, 0) 3

x 4

(0, â&#x2C6;&#x2019; 32(

1 (Because 2 x 3 is a 3 3 is not a common factor of N x and D x , x 2 vertical asymptote of f x . )

Vertical asymptote: x

Horizontal asymptote: y ÂªÂ¬Degree of N x

degree of D x ÂºÂ¼

NOT FOR SALE Section 2.6

x2 x 2 x 2x2 5x 6

40. f x

1 x 2

1 x 2 x 3

46. f x

(b) x-intercepts: 1, 0 , 2, 0

(b)

1Âˇ Â§ y-intercept: Â¨ 0, Â¸ 3Âš ÂŠ 2, x

Horizontal asymptote: y 4

f x

9 35

1, x

0 and

3 â&#x2C6;&#x2019;2

5 12

1 3

5 9

(c) Because there are only a finite number of pixels, the utility may not attempt to evaluate the function where it does not exist. 47. (a) Domain of f : all real numbers x except x

Domain of g: all real numbers x

4 3

, g x

195

â&#x2C6;&#x2019;2

x2 2x

(a) Domain of f: All real numbers x except x x 2 Domain of g: All real numbers x

(a) Domain: all real numbers x except x 1, x 2, and x 3

(d)

x 2 x 2

Ration Rational Functions

(b)

0, 2

(2, 0)

(â&#x2C6;&#x2019;1, 0)

â&#x2C6;&#x2019;3

x 2

(0, â&#x2C6;&#x2019; 13 (

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;4

(c) Because there are only finitely many pixels, the graphing utility may not attempt to evaluate the function where it does not exist.

â&#x2C6;&#x2019;5

41. g

48. f x

42. e 43. a

2x 6 , g x

x 2 7 x 12

2 x 4

(a) Domain of f: All real numbers x except x x 4

44. f 45. (a) Domain of f : all real numbers x except x

Domain of g: All real numbers x except x (b)

3 and 4

Domain of g: all real numbers x (b)

1 â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;1

2 â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;3

(c) Because there are only finitely many pixels, the graphing utility may not attempt to evaluate the function where it does not exist.

(c) Because there are only a finite number of pixels, the utility may not attempt to evaluate the function where it does not exist.

196

NOT FOR SALE

Chapter 2

Polynomial ynomial and Rational Functions

x2 9 x

49. h x

9 x

(a) Domain: all real numbers x except x

0 y

(b) x-intercepts: 3, 0 , 3, 0

y=x

h x

9 2

x2 5 x

50. g x

0 4

7 4

5 2

7 4

9 2

(a) Domain: all real numbers x except x

Slant asymptote: y

g x

14 3

â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;8

9 2

14 3

1 x

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;2

x 2

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;4

1 x2 x

1 x

(a) Domain: all real numbers x except x

(b) x-intercepts: 1, 0 , 1, 0

f x

â&#x2C6;&#x2019;6

52. f x

(a) Domain: all real numbers x except x

y=x

(b) No intercepts

(d)

x 8

â&#x2C6;&#x2019;4

2 x2 1 x

51. f x

5 x

(3, 0)

â&#x2C6;&#x2019;8 â&#x2C6;&#x2019; 6

(b) No intercepts

(d)

(â&#x2C6;&#x2019;3, 0)

Slant asymptote: y

33 4

9 2

33 4

73 6

(d)

f x

35 6

15 4

3 2

15 4

35 6

y = â&#x2C6;&#x2019;x

8 6

4 2 â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;4

y = 2x x

â&#x2C6;&#x2019;2

(â&#x2C6;&#x2019; 1, 0) â&#x2C6;&#x2019;8 â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;2

(1, 0) 4

x 6

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;8

NOT FOR SALE Section 2.6

x2 1 x

53. g x

1 x

(a) Domain: all real numbers x except x

(b) No intercepts (c) Vertical asymptote: x Slant asymptote: y (d)

x g x

17 4

t 5

197

26 t 5 5

0 x

5 2

17 4

37 6

t2 1 t 5

55. f t

Ration Rational Functions

t 5

Slant asymptote: y (d)

f t

1 5

6 4

y=x

2 â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;2

y=5â&#x2C6;&#x2019;t

(0, â&#x2C6;&#x2019; 15(

â&#x2C6;&#x2019;6

5 t

â&#x2C6;&#x2019; 20 â&#x2C6;&#x2019; 15 â&#x2C6;&#x2019;10 â&#x2C6;&#x2019; 5

x2 x 1

54. h x

1 x 1

x 1

(a) Domain: all real numbers x except x (b) Intercept: 0, 0

Slant asymptote: y

x 1

(d)

h x

16 5

4 3

x2 3x 1

56. f x

1 1 1 x 3 9 9 3 x 1

16 3

36 5

Slant asymptote: y

1 1 x 3 9

(d)

y=x+1

(0, 0) â&#x2C6;&#x2019;4

1 3

(b) Intercept: 0, 0

(a) Domain: all real numbers x except x

f x

9 8

4 5

1 2

1 2 1 2

1 2

4 7

x 2

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;4

y = 1x â&#x2C6;&#x2019; 1 3 9

2 3 1 3

â&#x2C6;&#x2019;1

(0, 0) x 1 3

2 3

4 3

198

NOT FOR SALE

Chapter 2

Polynomial ynomial and Rational Functions

x3 x 4

57. f x

4x x 4

(a) Domain: all real numbers x except x

Slant asymptote: y 6

f x

1 x 1

(b) y-intercept: 0, 1

(a) Domain: all real numbers x except x

(b) Intercept: 0, 0

(d)

x2 x 1 x 1

59. f x

27 4

16 3

1 3

16 3

27 4

(d)

x f x

21 5

1 x

7 3

13 3

y 8 8

y=x

4 2

(0, 0) 4

y=x

â&#x2C6;&#x2019; 8 â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;4

(0, â&#x2C6;&#x2019;1) â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;4

x3 2x 8

58. g x

1 4x x 2 2 2x 8

(a) Domain: all real numbers x except x

(d)

x g x

27 8

8 3

1 6

8 3

27 8

(d)

x f x

107 8

38 5

47 4

68 5

(0, 0) â&#x2C6;&#x2019; 8 â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;4

3 x 2

1 x 2

Slant asymptote: y

2x 1

(a) Domain: all real numbers x except x

(b) Intercept: 0, 0

2x2 5x 5 x 2

60. f x

x 4

y = 2x â&#x2C6;&#x2019; 1

y = 12 x

â&#x2C6;&#x2019;9 â&#x2C6;&#x2019;6 â&#x2C6;&#x2019; 3 0, â&#x2C6;&#x2019; 52

(

x 3

9 12 15

(

â&#x2C6;&#x2019;9

NOT FOR SALE Section 2.6

2 x

2 x3 x 2 2 x 1 x 2 3x 2

61. f x

2 x 2 3x 1 x 2

2x 7

1 x 1 x 1

x 1 x 2

15 , x 2

1 x 1

, x 2

y 18

â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;5 â&#x2C6;&#x2019;4 â&#x2C6;&#x2019; 3

(d)

45 2

f x

3 2

1 2

2x 7

9 , x 1

(1, 0) x

â&#x2C6;&#x2019;1

) 12 , 0)

â&#x2C6;&#x2019;12

â&#x2C6;&#x2019;18

y = 2x â&#x2C6;&#x2019; 7

â&#x2C6;&#x2019;24 â&#x2C6;&#x2019;30

â&#x2C6;&#x2019;36

x 2 x 2 2 x 1

x 2 x 1

2 x3 x 2 8 x 4 x 2 3x 2

62. f x

(0, 12 (

2x 7

Slant asymptote: y

x z 1

199

x z 1 1 and x

(a) Domain: all real numbers x except x

2 x

Ration Rational Functions

x z 2

(a) Domain: all real numbers x except x

1 and x

(b) y-intercept: 0, 2

y 30

Slant asymptote: y

2x 7

24 18

(d)

x f x

63. f x

5 4

1 2

3 2

35 2

x 2

2 x 3

1 1 2

x2 5x 8 x 3

Line: y

y = 2x + 7

(

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;6

(

â&#x2C6;&#x2019; 1, 0

(â&#x2C6;&#x2019;2, 0)

x 4

(0, â&#x2C6;&#x2019;2)

64. f x

2 x2 x x 1

2x 1

Slant asymptote: y 3

Line: y

x 2

1 x 1

Domain: all real numbers x except x

Vertical asymptote: x

Vertical asymptote: x Slant asymptote: y

2x 1

2x 1 6

â&#x2C6;&#x2019; 12

â&#x2C6;&#x2019; 14

â&#x2C6;&#x2019; 10

INSTRUCTOR USE ONLY â&#x2C6;&#x2019;8

200

Chapter 2

NOT FOR SALE

Polynomial ynomial and Rational Functions

1 3x 2 x3 x2

65. g x

1 3 x x2

Domain: all real numbers x except x Vertical asymptote: x Slant asymptote: y

x 3

1 x2

(a) x-intercepts: 1, 0 , 2, 0

x 3

Line: y

â&#x2C6;&#x2019; 12

12 â&#x2C6;&#x2019;4

12 2 x x 2 2 4 x

66. h x

1 2 x 1 2 4 x

Domain: all real numbers x except x

1 x 1 2

Line: y

(a)

x 1 x 3 x 1

0 0 1

2x x 3

68. y

1, x

1 x 2

25,000 p , 0 d p 100 100 p 300,000

â&#x2C6;&#x2019; 16

100 0

25,000 15

(b) C

100 15 25,000 50

x 1 x 3

(a) x-intercept: 1, 0

(b)

x 2 3x 2

â&#x2C6;&#x2019;6

67. y

71. C

2 x

Vertical asymptote: x Slant asymptote: y

x 3

(b) 0

x 3

2 x

x 3

70. y

| $4411.76

100 50 25,000 90

100 90

$25,000 $225,000

(c) C o f as x o 100. No, it would not be possible to supply bins to 100% of the residents because the model is undefined for p 100. 20 5 3t

,t t 0 1 0.04t

72. N

(a)

1400

(a) x-intercept: 0, 0

2x x 3 2x

(b) 0

69. y

1 x x

(a) x-intercepts: 1, 0 , 1, 0

(b)

0 x x2 x

n 0

200 0

(b) N 5 | 333 deer N 10

500 deer

N 25

800 deer

60 0.04

1500 deer

1 x x 1 x 1 r1

NOT FOR SALE Section 2.6

73. (a) A

Ration Rational Functions

201

xy and

x 4 y

30 30 x 4

y 2

y Thus, A

2 x 22 x 4

30 x 4

2 x x 11

Â§ 2 x 22 Âˇ xÂ¨ Â¸ ÂŠ x 4 Âš

x 4

(b) Domain: Since the margins on the left and right are each 2 inches, x ! 4. In interval notation, the domain is 4, f . (c)

200

40 4 0

The area is minimum when x | 11.75 inches and y | 5.87 inches.

y1 Area

160

102

69.143

69.333

70.909

The area is minimum when x is approximately 12. 74. A

xy and

3 y 2

64 x 3

y 2

y Thus, A

200

64 x 3

Â§ 2 x 58 Âˇ xÂ¨ Â¸ ÂŠ x 3 Âš

2 x 58 x 3

39 0

2 x x 29

, x ! 3. x 3

By graphing the area function, we see that A is minimum when x | 12.8 inches and y | 8.5 inches. 75. (a) Let t1 time from Akron to Columbus and t2 time from Columbus back to Akron.

xt1

100 Â&#x; t1

yt2

100 Â&#x; t2

50 t1 t2

(b) Vertical asymptote: x Horizontal asymptote: y (c)

100 x 100 y 200

25 25

200

65 0

(d)

150

87.5

66.7

56.3

45.8

42.9

t1 t2 100 100 x y 100 y 100 x 25 y 25 x 25 x

4 xy xy xy 25 y

(e) Sample answer: No. You might expect the average speed for the round trip to be the average of the average speeds for the two parts of the trip.

25 x

y x 25

(f ) No. At 20 miles per hour you would use more time in one direction than is required for the round trip at an average speed of 50 miles per hour.

4 4

25 x . x 25

INSTRUCTOR USE ONLY Thus, y

202

NOT FOR SALE

Chapter 2

Polynomial ynomial and Rational Functions

76. Yes; No; Every rational function is the ratio of two N x

polynomial functions of the form f x

. D x

80. (a) True. When x

asymptote, therefore D 1

(b) True. Since the graph of f has a horizontal asymptote at y

77. False. Polynomial functions do not have vertical asymptotes.

2, the degrees of N x and

D x are equal. (c) False. Since the horizontal asymtptote is at y 2, which shows that the ratio of the leading

x crosses x 1 0, which is a horizontal asymptote.

78. False. The graph of f x

1 the graph of f has a vertical

coefficients of N x and D x is 2, not 1.

79. False. A graph can have a vertical asymptote and a horizontal asymptote or a vertical asymptote and a slant asymptote, but a graph cannot have both a horizontal asymptote and a slant asymptote.

81. b 82. c

A horizontal asymptote occurs when the degree of N x

is equal to the degree of D x or when the degree of

N x is less than the degree of D x . A slant asymptote occurs when the degree of N x is greater than the degree of D x by one. Because the degree of a polynomial is constant, it is impossible to have both relationships at the same time.

Section 2.7 Nonlinear Inequalities 1. positive; negative

3. zeros; undefined values

2. key; test intervals

4. P

R C

5. x 2 3 0

(a) x

3 2

(b) ?

3 0

0 2

3 2

3 0

6 0 No, x

(c)

3 0

3 is not

Yes, x

a solution.

(d)

22 0

34 0

0 is

3 2

Yes, x

a solution.

3 0

5 is not

No, x

a solution.

6. x 2 x 12 t 0

(a) x

5 2

(b)

5 ?

5 12 t 0

0 2

0 0 12 t 0

8 t 0 Yes, x

5 is

a solution.

12 t 0 No, x

0 is not

(d) ?

4 2

4 12 t 0

3 2

3 12 t 0 ?

9 3 12 t 0

16 4 12 t 0

0 t 0

8 t 0

a solution.

Yes, x

4 is

a solution.

Yes, x

3 is

a solution.

NOT FOR SALE Section 2.7

(b)

Yes, x

5 is

a solution.

(c)

4 2 ? t 3 4 4 6 is undefined. 0 No, x 4 is not

5 2 ? t 3 5 4 7 t 3

9 2

(d)

9 2

9 2 ? 2 t 3 9 4 2 5 t 3 17 9 No, x is not 2 a solution.

9 2 ? 2 t 3 9 4 2 13 t 3

a solution.

9 is 2 a solution. Yes, x

3x 2 1 x2 4

(a) x

3 2

3 1

(b)

(c)

12 1 8 No, x 2 is not

Yes, x

a solution.

9. 3x 2 x 2 3x 2 x 1

3 x

2 x 1

0 Â&#x; x

3 1 5 1 is

0 3 0

10. 9 x3 25 x 2

x 2 9 x 25

0 1 Yes, x

9 x 25

0 Â&#x; x

25 9

The key numbers are 0 and 1 1 x 5

25 . 9

1 1 x 5

x 4

x 5 x 4 x 5 0 Â&#x; x 4

x 5

0 Â&#x; x

27 1 13 No, x 3 is not

0 is

a solution.

a solution. 12.

2 x x 2 x 1

x x 1 2 x 2

2 x 1

x x 2x 4 x 2 x 1

x x x

0 Â&#x; x

3 3 3

(d)

The key numbers are 23 and 1.

11.

203

x 2 t 3 x 4

(a) x

Nonlinear Inequalities

4 x 1

2 x 1

x 4

0 Â&#x; x

x 1

0 Â&#x; x

2 x 1

x 2

0 Â&#x; x

x 1

0 Â&#x; x

The key numbers are 2, 1, 1, and 4.

The key numbers are 4 and 5.

204

Chapter 2

NOT FOR SALE

Polynomial ynomial and Rational Functions x2 9

13.

15.

x2 9 0

x 2 4 x 4 d 25

3 x 3 0

Key numbers: x

2 d 25

x 2 4 x 21 d 0

7 x 3 d 0

Test intervals: f, 3 , 3, 3 , 3, f

Key numbers: x

Test: Is x 3 x 3 0?

Test intervals: f, 7 , 7, 3 , 3, f

Conclusion

Test: Is x 7 x 3 d 0?

Positive

Interval

â&#x20AC;&#x201C;9

Negative

Positive

x-Value

Value of x 2 9

f, 3

â&#x20AC;&#x201C;4

3, 3

3, f

Interval

7, x

Solution set: 3, 3

x-Value

Value of 7 x 3

f, 7

â&#x20AC;&#x201C;8

1 11

7, 3

7 3

3, f

11 1

Conclusion

Positive

Negative

Positive

Solution set: > 7, 3@ x 2 d 16

14.

x 2 16 d 0

4 x 4 d 0

Key numbers: x

16.

3 t 1 2

x2 6x 8 t 0

Test intervals: f, 4 , 4, 4 , 4, f

Test: Is x 4 x 4 d 0?

Key numbers: x

Interval

x-Value Value of x 2 16 â&#x20AC;&#x201C;5

Positive

4, 4

Negative

4, f

Positive

2, x

Test intervals: f, 2 Â&#x; x 2 x 4 ! 0

Conclusion

f, 4

2 x 4 t 0

2, 4 Â&#x; x 2 x 4 0 4, f Â&#x; x 2 x 4 ! 0 Solution set: f, 2@ Â&#x2030; >4, f

Solution set: > 4, 4@ 17.

x2 4x 4 t 9 x2 4 x 5 t 0

5 x 1 t 0

Key numbers: x

5, x

Test intervals: f, 5 , 5, 1 , 1, f

Test: Is x 5 x 1 t 0? Interval

f, 5

x-Value

Value of x 5 x 1

1 7

5, 1

5 1

1, f

7 1

Conclusion

Positive

Negative

Positive

Solution set: f, 5@ Â&#x2030; >1, f

NOT FOR SALE Section 2.7

18.

x 2 6 x 9 16

21.

x 1 x 7 0 Key numbers: x

Interval

Solution set: 1, 7

f, 3

x2 x 6

Value of 3 x 1

Conclusion

1 5

Positive

3, 1

3 1

Negative

1, f

5 1

Positive

3 x 2 0

Key numbers: x

3, x

x2 2x 8 ! 0

Test: Is x 3 x 2 0? Interval

f, 3

x-Value

Value of x 3 x 2

1 6

3, 2

3 2

2, f

6 1

x2 ! 2 x 8

22.

Test intervals: f, 3 , 3, 2 , 2, f

Conclusion

4 x 2 ! 0

Key numbers: x

2, x

Test intervals: f, 2 , 2, 4 , 4, f

Positive

Negative

Positive

Solution set: 3, 2

x2 2 x ! 3

Test: Is x 4 x 2 ! 0? Interval

x-Value

Value of 4 x 2

Conclusion

f, 2

â&#x20AC;&#x201C;3

7 1

Positive

2, 4

4 2

Negative

4, f

1 7

Positive

Solution set: f, 2 Â&#x2030; 4, f

x2 2 x 3 ! 0

x-Value

Solution set: 3, 1

x2 x 6 0

20.

Test: Is x 3 x 1 0?

1, 7 Â&#x; x 1 x 7 0 7, f Â&#x; x 1 x 7 ! 0

3, x

Test intervals: f, 3 , 3, 1 , 1, f

Test intervals: f, 1 Â&#x; x 1 x 7 ! 0

19.

3 x 1 0

Key numbers: x 1, x

205

x2 2 x 3 0

x2 6 x 7 0

Nonlinear Inequalities

3 x 1 ! 0

Key numbers: x

3, x

Test intervals: f, 3 Â&#x; x 3 x 1 ! 0

3, 1 Â&#x; x 3 x 1 0 1, f Â&#x; x 3 x 1 ! 0 Solution set: f, 3 Â&#x2030; 1, f

206

NOT FOR SALE

Chapter 2

Polynomial ynomial and Rational Functions

3 x 2 11x ! 20

23.

25.

3 x 2 11x 20 ! 0

3x 4 x 5 ! 0 Key numbers: x

x 2 3x 18 ! 0 3 x 6 ! 0

Key numbers: x

5, x

3, x

Test intervals: f, 3 , 3, 6 , 6, f

Test intervals: f, 54 , 34 , 5 , 5, f

Test: Is x 3 x 6 ! 0?

Test: Is 3x 4 x 5 ! 0?

Interval

x-Value

f, 43

43 , 5

3 x

Value of 4 x 5

Conclusion

â&#x20AC;&#x201C;3

5 8

Positive

4 â&#x20AC;&#x201C; 5

Negative

22 1

Positive

5, f

x-Value

Value of 3 x 6

f, 3

â&#x20AC;&#x201C;4

1 10

3, 6

3 â&#x20AC;&#x201C; 6

6, f

10 1

Conclusion

Positive

Negative

Positive

Solution set: f, 3 Â&#x2030; 6, f

Solution set: f, 43 Â&#x2030; 5, f

26.

x 2 x 2 4 x 2 d 0

24. 2 x 2 6 x 15 d 0 2 x 2 6 x 15 t 0

6 r 6r

6 4 2 15

2 2

2 x 2 4 d 0

2 x 2 d 0 2

2, x

Test intervals: f, 2 Â&#x; x3 2 x 2 4 x 8 0

156

2, 2 Â&#x; x3 2 x 2 4 x 8 0 2, f Â&#x; x3 2 x 2 4 x 8 ! 0

6 r 2 39 4

Solution set: f, 2@

39 2

Key numbers: x

3 r 2

x3 2 x 2 4 x 8 d 0

3 2

39 ,x 2

3 2

39 2

39 Â· 2 Â¸ Â&#x; 2 x 6 x 15 0 2 Â¸Â¹

Â§3 Â¨Â¨ Â©2

39 3 , 2 2

39 Â· 2 Â¸ Â&#x; 2 x 6 x 15 ! 0 2 Â¸Â¹

Â§3 Â¨Â¨ Â©2

39 Â· , f Â¸Â¸ Â&#x; 2 x 2 6 x 15 0 2 Â¹

39 Â· , f Â¸Â¸ 2 Â¹

NOT FOR SALE Section 2.7

Nonlinear Inequalities

207

x3 3 x 2 x ! 3

27.

x3 3x 2 x 3 ! 0 x 2 x 3 x 3 ! 0

x 3 x 2 x 3 x 1 x

1 ! 0 1 ! 0

Key numbers: x

1, x

Test intervals: f, 1 , 1, 1 , 1, 3 , 3, f

Test: Is x 3 x 1 x 1 ! 0? Interval

x-Value

Value of x 3 x 1 x 1

Conclusion Negative

f, 1

â&#x20AC;&#x201C;2

5 1 3

1, 1

3 1 1

1, 3

1 3 1

3, f

1 5 3

15 3

Positive

Negative

Positive

Solution set: 1, 1 Â&#x2030; 3, f

28.

2 x3 13 x 2 8 x 46 t 6

29.

2 x 2 2 x 3 0

2 x3 13 x 2 8 x 52 t 0 x 2 2 x 13 4 2 x 13 t 0

2 x

2 x 13 x 2 13 x 2 x

Key numbers: x

4 t 0

0, 32 Â&#x; 2 Â&#x; 2 x 2 x 3 0 32 , f Â&#x; 2 x 2 x 3 ! 0 Solution set: f, 0 Â&#x2030; 0, 32

2, x

Test intervals:

f, 132 Â&#x; 2 x 132 , 2 Â&#x; 2 x

13x 2 8 x 52 0

13 x 2 8 x 52 ! 0

2, 2 Â&#x; 2 x3 13x 2 8 x 52 0 2, f Â&#x; 2 x3 13x 2 8 x 52 ! 0 , 2ÂºÂ¼, >2, f

Solution set: ÂªÂ¬ 13 2

3 2

0, x

Test intervals: f, 0 Â&#x; 2 x 2 2 x 3 0

2 t 0

13 , 2

4 x3 6 x 2 0

3 2 x â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;1

30. 4 x3 12 x 2 ! 0 4 x 2 x 3 ! 0

Key numbers: x

0, x

Test intervals: f, 0 Â&#x; 4 x 2 x 3 0

0, 3 Â&#x; 4 x 2 x 3 0 3, f Â&#x; 4 x 2 x 3 ! 0 Solution set: 3, f

x 1

208

Chapter 2

Polynomial ynomial and Rational Functions

x3 4 x t 0

31.

33.

x x 2 x 2 t 0

Key numbers: x

2, 1

1, f

Â&#x; x 1 x 2 ! 0 2

Solution set: > 2, f

x â&#x2C6;&#x2019;4

x 2

1, x

Solution set: > 2, 0@ Â&#x2030; >2, f

Test intervals: f, 2 Â&#x; x 1 x 2 0

2, 0 Â&#x; x x 2 x 2 ! 0 0, 2 Â&#x; x x 2 x 2 0 2, f Â&#x; x x 2 x 2 ! 0

Key numbers: x r2

0, x

Test intervals: f, 2 Â&#x; x x 2 x 2 0

â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;1

1 x 2 t 0

â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;1

34. x 4 x 3 d 0

Key numbers: x

32. 2 x3 x 4 d 0

0, x

Test intervals: f, 0 Â&#x; x 4 x 3 0

x 3 2 x d 0

Key numbers: x

0, x

0, 3 Â&#x; x 4 x 3 0 3, f Â&#x; x 4 x 3 ! 0

Test intervals: f, 0 Â&#x; x3 2 x 0

Solution set: f, 3@

0, 2 Â&#x; x3 2 x ! 0 2, f Â&#x; x3 2 x 0

x 1

Solution set: f, 0@ Â&#x2030; >2, f

x â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;1

35. 4 x 2 4 x 1 d 0

(2 x 1) 2 d 0 Key number: x

1 2

Test Interval

x-Value

1Âş Â§ Â¨ f, Âť 2Âź ÂŠ

Â§1 Âˇ Â¨ , fÂ¸ ÂŠ2 Âš

Polynomial Value

Conclusion

[2(0) 1]2

Positive

[2(1) 1]2

Positive

The solution set consists of the single real number

1 . 2

36. x 2 3 x 8 ! 0

Using the Quadratic Formula you can determine the key numbers are x Test Interval

x-Value

Polynomial Value

( f, f)

(0) 2 3(0) 8

3 r 2

23 i. 2

Conclusion 8

Positive

The solution set is the set of all real numbers.

NOT FOR SALE Section 2.7

Nonlinear Inequalities

209

37. x 2 6 x 12 d 0

Using the Quadratic Formula, you can determine that the key numbers are x Test Interval ( f, f)

x-Value

Polynomial Value

(0) 6(0) 12 2

3i .

Conclusion 12

Positive

The solution set is empty, that is there are no real solutions. 38. x 2 8 x 16 ! 0

( x 4) 2 ! 0 Key number: x

Test Interval

x-Value

Polynomial Value

Conclusion

( f, 4)

(0 4)

Positive

(4, f)

(5 4) 2

Positive

The solution set consists of all real numbers except x

39.

4x 1 ! 0 x

40.

Key numbers: x

1 4

0, x

14 , 14 , f

Test intervals: f, 0 , 0, Test: Is

4, or ( f, 4) Â&#x2030; (4, f).

Key numbers: x

x-Value

f, 0

â&#x20AC;&#x201C;1

1 8

Interval 4x 1 x

Value of 5 1 1 2 1 8

3 1

x-Value

Conclusion Positive

1, x

0, x

Test intervals: f, 1 , 1, 0 , 0, 1 , 1, f

4x 1 ! 0? x

Interval

x2 1 0 x x 1 x 1 0 x

Negative

Positive

Value of 1 x 1

Conclusion

f, 1

â&#x20AC;&#x201C;2

1, 0

1 2

0, 1

1 2

1, f

3 1

1 3

3 2

Negative

3 2

Positive

3 2

Negative

Positive

Solution set: f, 1 Â&#x2030; 0, 1

210

41.

NOT FOR SALE

Chapter 2

Polynomial ynomial and Rational Functions

3x 5 t 0 x 5

43.

Key numbers: x

5 ,x 3

5Â· Â§5 Â· Â§ Test intervals: Â¨ f, Â¸, Â¨ , 5 Â¸, 5, f

Key numbers: x

3x 5 t 0? x 5

x-Value

Interval

Value of

3x 5 x 5

6 5 2 5

1 3

Negative

5, f

18 5 6 5

Positive

5Âº Â§ Solution set: Â¨ f, Â» Â&#x2030; 5, f

3Â¼ Â©

42.

5 7x d 4 1 2x 5 7 x 4 1 2 x

1 2x

x 12 3 t 0 x 2 x 12 3 x 2

t 0 x 2 6 2x t 0 x 2

Key numbers: x

2, x

6 2x 0 x 2 6 2x ! 0 2, 3 Â&#x; x 2 6 2x 0 3, f Â&#x; x 2

Test intervals: f, 2 Â&#x;

d 0

1 ,x 2

Solution interval: 2, 3@

1Â· Â§ 1 Â· Â§ Test intervals: Â¨ f, Â¸, Â¨ , 1Â¸, 1, f

Solution set: f, 1 Â&#x2030; 4, f

44.

1 x d 0 1 2x Key numbers: x

4 x 0 x 1 4 x ! 0 1, 4 Â&#x; x 1 4 x 0 4, f Â&#x; x 1

Conclusion

5 5

5Â· Â¸ 3Â¹

1, x

Test intervals: f, 1 Â&#x;

Â§ Â¨ f, Â©

x 6 2 0 x 1 x 6 2 x 1

0 x 1 4 x 0 x 1

1 x d 0? 1 2x

Interval

x-Value

Value of

1 x 1 2x

Conclusion

Negative

Positive

â&#x20AC;&#x201C;1

2 1

1 1

1, f

1 5

Negative

1Â· Â§ Solution set: Â¨ f, Â¸ Â&#x2030; >1, f

2Â¹ Â©

NOT FOR SALE Section 2.7

45.

2 1 ! x 5 x 3 2 1 ! 0 x 5 x 3 2 x 3 1 x 5

! 0 x 5 x 3

47.

x 11 ! 0 x 5 x 3

Key numbers: x

3, x

x 11 ! 0 x 5 x 3

Â&#x;

30 5 x Â§ 3 Â· 0 Â¨ , 3Â¸ Â&#x; x 3 4 x 3

Â© 4 Â¹

x 11 ! 0 5 x 3

3 ,x 4

3, x

3Â· 30 5 x Â§ Test intervals: Â¨ f, Â¸ Â&#x; ! 0 4Â¹ x 3 4 x 3

Â©

x 11 0 3, 11 Â&#x; x 5 x 3

11, f

1 9 d x 3 4x 3 1 9 d 0 x 3 4x 3 4 x 3 9 x 3

d 0 x 3 4 x 3

Key numbers: x

x 11 0 Test intervals: f, 5 Â&#x; x 5 x 3

Â&#x;

211

30 5 x d 0 x 3 4 x 3

5, x

5, 3

Nonlinear Inequalities

Solution set: 5, 3 Â&#x2030; 11, f

3, 6

Â&#x;

6, f

Â&#x;

30 5 x ! 0 3 4 x 3

30 5 x 0 3 4 x 3

Â§ 3 Â· Solution set: Â¨ , 3Â¸ Â&#x2030; >6, f

Â© 4 Â¹ 46.

5 3 ! x 6 x 2 5 x 2 3 x 6

! 0 x 6 x 2

48.

2 x 28 ! 0 6 x 2

Key numbers: x

14, x

2, x

2 x 28 0 Test intervals: f, 14 Â&#x; x 6 x 2

14, 2

2, 6

6, f

Â&#x;

2 x 28 ! 0 x 6 x 2

Â&#x;

x x

2 x 28 0 6 x 2

2 x 28 ! 0 6 x 2

Solution intervals: 14, 2 Â&#x2030; 6, f

1 1 t x x 3 1 x 3 1 x

t 0 x x 3

3 t 0 x x 3

Key numbers: x

3, x

Test intervals: f, 3 Â&#x;

3, 0

0, f

Â&#x; Â&#x;

3 ! 0 x x 3

3 0 x x 3

3 ! 0 x x 3

Solution intervals: f, 3 Â&#x2030; 0, f

212

49.

Chapter 2

Polynomial ynomial and Rational Functions

x2 2x d 0 x2 9 x x 2

d 0 x 3 x 3

Key numbers: x

51.

2, x

0, x

Test intervals: f, 3 Â&#x;

3, 2

2, 0

Â&#x;

0, 3

Â&#x;

3, f

Â&#x;

x x

3x 2 x 2 ! 0 x 1 x 1

x x 2

! 0

3 x 3

x x 2

3 x 3

x x 2

3 2x ! 1 x 1 x 1 3 x 1 2 x x 1 1 x 1 x 1

! 0 x 1 x 1

3 x 3

x x 2

3 x 3

Key numbers: x

! 0

3x 2 x 2 ! 0 x 1 x 1

1, 1

Â&#x;

3x 2 x 2 0 x 1 x 1

1, f

Â&#x;

3x 2 x 2 ! 0 x 1 x 1

Solution set: f, 1 Â&#x2030; 1, f

! 0

3 x 3

Test intervals: f, 1 Â&#x;

x x 2

1, x

Solution set: 3, 2@ Â&#x2030; >0, 3

52.

50.

x2 x 6 t 0 x x 3 x 2 t 0 x Key numbers: x

x 3x d 3 x 1 x 4 3x x 4 x x 1 3 x 4 x 1

d 0 x 1 x 4

x 2 4 x 12 d 0 x 1 x 4

3, x

0, x

Test intervals: f, 3 Â&#x;

3, 0

Â&#x;

0, 2

Â&#x;

2, f

Â&#x;

x 6 x 2

3 x 2

x x 3 x 2

x x x 3

x x 3 x 2

Solution set: > 3, 0 Â&#x2030; >2, f

! 0

0 ! 0

Key numbers: x

4, x

Test intervals: f, 4 Â&#x;

1 x 4

2, x

1, x

d 0 6

x 6 x 2

x 1 x 4

x 6 x 2

! 0 4, 2 Â&#x; x 1 x 4

x 6 x 2

0 2, 1 Â&#x; x 1 x 4

x 6 x 2

! 0 1, 6 Â&#x; x 1 x 4

x 6 x 2

0 6, f Â&#x; x 1 x 4

Solution set: f, 4 Â&#x2030; > 2, 1 Â&#x2030; >6, f

NOT FOR SALE Section 2.7

53. y

x2 2x 3

58. y

(a) y d 0 when x d 1 or x t 3.

54. y

2 x 2

x 1

(b) y t 8 when 2 d x 1.

2x 1 59. y

(a) y d 0 when 2

213

(a) y d 0 when 1 x d 2.

(b) y t 3 when 0 d x d 2. 1 x2 2

Nonlinear Inequalities

2 d x d 2

2 x2 x 4 2

(a) y t 1 when x d 2 or x t 2.

(b) y t 7 when x d 2 or x t 6.

This can also be expressed as x t 2. 55. y

1 x3 8

1x 2

(b) y d 2 for all real numbers x. This can also be expressed as f x f. 60. y

5x x2 4

(a) y t 1 when 1 d x d 4. (b) y d 0 when f x d 0.

(a) y t 0 when 2 d x d 0 or 2 d x f. (b) y d 6 when x d 4. 56. y

x3 x 2 16 x 16 4 x2 t 0

61.

x 2 x t 0

Key numbers: x

Test intervals: f, 2 Â&#x; 4 x 2 0

2, 2 Â&#x; 4 x 2 ! 0 2, f Â&#x; 4 x 2 0

(a) y d 0 when f x d 4 or 1 d x d 4. (b) y t 36 when x 57. y

2 or 5 d x f.

3x x 2

Domain: > 2, 2@ x2 4 t 0

62.

2 x 2 t 0

Key numbers: x

2, x

Test intervals: f, 2 Â&#x; x 2 x 2 ! 0 (a) y d 0 when 0 d x 2.

2, 2 Â&#x; x 2 x 2 0 2, f Â&#x; x 2 x 2 ! 0

INSTRUCTOR USE ONLY (b) y t 6 when 2 x d 4.

f, 2@ Â&#x2030; >2, f

Domain: f,

214

63.

NOT FOR SALE

Chapter 2

Polynomial ynomial and Rational Functions

x 2 9 x 20 t 0

4 x 5 t 0

Key numbers: x

4, x

x-Value

f, 4

4 5

4, 5

9 2

5, f

Conclusion

12 12

3, 0

Negative

Positive

Domain: f, 4@ Â&#x2030; >5, f

2 x 9 2 x t 0

67.

9 r 2

Key numbers: x

x-Value

0, 3

Â&#x;

3, f

Â&#x;

x 0 3 x 3

x ! 0 3 x 3

x 0 3 x 3

x ! 0 3 x 3

Key numbers: x | r 3.51

Value of Conclusion 9 2 x 9 2 x

19 1

0.4 x 2 5.26 10.2 0.4 x 2 12.35 0

Test intervals: f, 3.51 , 3.51, 3.51 , 3.51, f

Solution set: 3.51, 3.51

Negative 68. 1.3 x 2 3.78 ! 2.12

9 9

1 19

Positive

1.3 x 2 1.66 ! 0 Key numbers: x | r1.13

Negative

69. 0.5 x 2 12.5 x 1.6 ! 0

x t 0 x 2 2 x 35 x t 0 x 5 x 7

Key numbers: x

Test intervals: f, 1.13 , 1.13, 1.13 , 1.13, f

Solution set: 1.13, 1.13

Âª 9 9Âº Domain: Â« , Â» Â¬ 2 2Â¼ 65.

Â&#x;

0.4 x 2 4.94 0

0, x

Domain: 3, 0@ Â&#x2030; 3, f

81 4 x 2 t 0

64.

3, x

Test intervals: f, 3 Â&#x;

Positive

2 1

x t 0 3 x 3

Key numbers: x

Value of 4 x 5

Test intervals: f, 4 , 4, 5 , 5, f

Interval

x t 0 x 9

66.

Key numbers: x | 0.13, x | 25.13 Test intervals: f, 0.13 , 0.13, 25.13 , 25.13, f

Solution set: 0.13, 25.13

5, x

0, x

Test intervals: f, 5 Â&#x;

5, 0

Â&#x;

7, f

Â&#x;

5 x 7

x ! 0 5 x

x 7

Â&#x;

0, 7

5 x 7

x ! 0 5 x 7

70. 1.2 x 2 4.8 x 3.1 5.3 1.2 x 2 4.8 x 2.2 0

Key numbers: x | 4.42, x | 0.42 Test intervals: f, 4.42 , 4.42, 0.42 , 0.42, f

Solution set: 4.42, 0.42

Domain: 5, 0@ Â&#x2030; 7, f

NOT FOR SALE Section 2.7

1 ! 3.4 2.3x 5.2

71.

16t 2 v0t s0

74. s

1 3.4 ! 0 2.3x 5.2 1 3.4 2.3 x 5.2

! 0 2.3x 5.2 7.82 x 18.68 ! 0 2.3 x 5.2

(a) 16t 2 128t

16t t 8

0, t

4 2 2

2, 4 2 2 ,

Solution set: 0 seconds d t 4 2 2 seconds and 4 2 2 seconds t d 8 seconds 75. 2 L 2W

100 Â&#x; W

50 L

LW t 500 L 50 L t 500 L 50 L 500 t 0 2

By the Quadratic Formula you have:

16t 2 160t

Key numbers: L

25 r 5 5

Test: Is L 50 L 500 t 0? 2

Solution set: 25 5 5 d L d 25 5 5 13.8 meters d L d 36.2 meters

It will be back on the ground in 10 seconds.

76. 2 L 2W

440 Â&#x; W

220 L

LW t 8000

16t 160t ! 384 2

(b)

4 2 2, t

f, 4 2 2 , 4 2 4 2 2, f

Solution set: 1.19, 1.30

0 Â&#x; t

Test intervals:

23.46 17.98 x 0 3.1x 3.7 23.46 17.98 x ! 0 1.19, 1.30 Â&#x; 3.1x 3.7 23.46 17.98 x 0 1.30, f Â&#x; 3.1x 3.7

t 8

Key numbers: t

Test intervals: f, 1.19 Â&#x;

16t t 10

t 2 8t 8 ! 0

Key numbers: x | 1.19, x | 1.30

0 Â&#x; t

16 t 2 8t 8 0

2 ! 5.8 3.1x 3.7 2 5.8 3.1x 3.7

! 0 3.1x 3.7 23.46 17.98 x ! 0 3.1x 3.7

(a) 16t 2 160t

16t

16t 2 128t 128 0

Solution set: 2.26, 2.39

16t 2 v0t s0

16t 2 128t

16t 2 128t 128

(b)

Test intervals: f, 2.26 , 2.26, 2.39 , 2.39, f

73. s

215

It will be back on the ground in 8 seconds.

Key numbers: x | 2.39, x | 2.26

72.

Nonlinear Inequalities

L 220 L t 8000

16t 160t 384 ! 0 2

L 220 L 8000 t 0

16 t 2 10t 24 ! 0

t 2 10t 24 0

By the Quadratic Formula we have:

Key numbers: L

4 t 6 0

110 r 10 41

Test: Is L 220 L 8000 t 0? 2

Key numbers: t

4, t

Test intervals: f, 4 , 4, 6 , 6, f

Solution set: 110 10 41 d L d 110 10 41 45.97 feet d L d 174.03 feet

Solution set: 4 seconds t 6 seconds

216

Chapter 2

NOT FOR SALE

Polynomial ynomial and Rational Functions

x 75 0.0005 x and C

77. R

30 x 250,000

R C

78. R

x 50 0.0002 x and C

12 x 150,000

R C

75x 0.0005 x 30 x 250,000

50 x 0.0002 x 2 12 x 150,000

0.0005 x 2 45 x 250,000

0.0002 x 2 38 x 150,000

P t 750,000

P t 1,650,000

0.0005 x 45 x 250,000 t 750,000

0.0002 x 2 38 x 150,000 t 1,650,000

0.0005 x 2 45 x 1,000,000 t 0

0.0002 x 2 38 x 1,800,000 t 0

Key numbers: x

40,000, x

50,000

(These were obtained by using the Quadratic Formula.) Test intervals: 0, 40,000 , 40,000, 50,000 , 50,000, f

The solution set is >40,000, 50,000@ or 40,000 d x d 50,000. The price per unit is R x

For x

75 0.0005 x. 40,000, p

$55. For x

50,000,

$50. So, for 40,000 d x d 50,000,

90,000 and x

100,000

Test intervals: 0, 90,000 , 90,000, 100,000 , 100,000, f

The solution set is >90,000, 100,000@ or 90,000 d x d 100,000. The price per unit is R p 50 0.0002 x. x 90,000, p $32. For x 100,000, For x

p $30. So, for 90,000 d x d 100,000, $30 d p d $32.

$50.00 d p d $55.00. 79. (a)

(b) N

0.00406t 4 0.0564t 3 0.147t 2 0.86t 72.2

(c)

The model fits the data well. (d) Using the zoom and trace features, the number of students enrolled in schools exceeded 74 million in the year 2001. (e) No. The model can be used to predict enrollments for years close to those in its domain but when you project too far into the future, the numbers predicted by the model increase too rapidly to be considered reasonable.

NOT FOR SALE Section 2.7

80. (a)

Load 2223.9 5593.9 10,312 16,378 23,792

1 R

81.

Nonlinear Inequalities

1 1 R1 2

2 R1

2 R RR1

2 R1

R 2 R1

2 R1 2 R1

217

Because R t 1, 2 R1 t1 2 R1 2 R1 1t 0 2 R1

2000 d 168.5d 2 472.1

(b)

R1 2 t 0. 2 R1

2472.1 d 168.5d 2 14.67 d d 2

Because R1 ! 0, the only key number is R1 2. The inequality is satisfied when R1 t 2 ohms.

3.83 d d

The minimum depth is 3.83 inches. 82. (a)

(b) The model fits the data well. (c)

42.16 0.236t 1 0.026t 42.16 0.236t 60 d 1 0.026t 42.16 0.236t 60 0 d 1 0.026t 17.84 1.324t 0 d 1 0.026t S

Key numbers: t | 38.5 and t | 13.5 Test Intervals

t-Value

0, 13.5

13.5, 38.5

38.5, f

Expression Value

Conclusion

17.0

Negative

18.0

Positive

878.0

Negative

So, the mean salary for classroom teachers will exceed $60,000 during the year 2013. (d) No. The model yields negative values for values of t t 38.5. The graph also has a vertical asymptote at 500 t | 38.5. 13 After testing the intervals, you can see that the inequality is satisfied on the open interval (13.5, 38.5). 83. True.

84. True.

x 2 x 11x 12 3

3 x 1 x 4

The y-values are greater than zero for all values of x.

The test intervals are f, 3 , 3, 1 , 1, 4 , and

4, f .

218

Chapter 2

NOT FOR SALE

Polynomial ynomial and Rational Functions

85. x 2 bx 4

88. 2 x 2 bx 5

(a) To have at least one real solution, b 2 4ac t 0.

(a) To have at least one real solution, b 4ac t 0. 2

b 2 4 1 4 t 0

b 2 4 2 5 t 0

b 2 16 t 0

b 2 40 t 0

Key numbers: b

4, b

Key numbers: b

4, 4 Â&#x; b 4, f Â&#x; b 2

2 10

Test intervals: f, 2 10 Â&#x; b 2 40 ! 0

16 0

16 ! 0

Solution set: f, 4@ Â&#x2030; >4, f@ (b) b 2 4ac t 0 Key numbers: b

2 10, b

2 10, 2 10 Â&#x; b 40 0 2 10, f Â&#x; b 40 ! 0 Solution set: f, 2 10 ÂşÂź Â&#x2030; ÂŞÂŹ2 10, f

Test intervals: f, 4 Â&#x; b 16 ! 0 2

2 ac , b

(b) b 2 4ac t 0

2 ac

Similar to part (a), if a ! 0 and c ! 0,

b d 2 ac or b t 2ac.

86. x 2 bx 4

For part (b), the y-values that are less than or equal to 0 occur only at x 1.

89.

(a) To have at least one real solution, b 2 4ac t 0.

b 2 4 1 4 t 0 b 2 16 t 0 Key numbers: none

For part (c), there are no y-values that are less than 0.

Test intervals: f, f Â&#x; b 2 16 ! 0 Solution set: f, f

(b) b 2 4ac t 0 Similar to part (a), if a ! 0 and c 0, b can be any real number. 87. 3x 2 bx 10

For part (d), the y-values that are greater than 0 occur for all values of x except 2

(a) To have at least one real solution, b 2 4ac t 0.

b 2 4 3 10 t 0

90. (a) x

b 120 t 0

a, x

Key numbers: b

(b)

2 30, b

2 30

2 30, 2 30 Â&#x; b 120 0 2 30, f Â&#x; b 120 ! 0

Test intervals: f, 2 30 Â&#x; b 2 120 ! 0 2

Solution set: f, 2 30 ÂşÂź Â&#x2030; ÂŞÂŹ2 30, fÂşÂź (b) b 2 4ac t 0 Similar to part (a), if a ! 0 and c ! 0,

b d 2 ac or b t 2ac.

Review Exercises ffor Chapter 2

219

Review Exercises for Chapter 2 1. (a) y

3. g x

2x2

x2 2x x2 2x 1 1

Vertical stretch

1 1 2

Vertex: 1, 1

3 2

Axis of symmetry: x x

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;2

x-intercepts: 0, 0 , 2, 0

â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;4

(b) y

x x 2

x 2x 2

x 2 2

7 6

Vertical shift two units upward

5 4

4 3 x

â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1

1 1

4. f x

â&#x2C6;&#x2019;3

x 4 2 4, 6

Vertex:

Horizontal shift three units to the right

4 3 2 1 x 1

x 4

r 6

4 r

â&#x2C6;&#x2019;3

4 6

x-intercepts: 4 r

â&#x2C6;&#x2019;2

1 x2 2

Axis of symmetry: x

(b) y

x 2 8 x 16 16 10

â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019;1

x 2 8 x 10

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;2

2. (a) y

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;8

â&#x2C6;&#x2019;4

2 â&#x2C6;&#x2019;2

6 6, 0

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;6

Vertical shrink each y -value is multiplied by

1 2

and a vertical shift one unit downward y

4 3 2 1 x

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019; 3 â&#x2C6;&#x2019; 2

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;4

220

NOT FOR SALE

Chapter 2

5. h x

Polynomial ynomial and Rational Functions

7. h x

3 4 x x2

4 x 2 4 x 13

x 2 4 x 3

4 x 2 x 13 1 14 13 4 4 x x 14 1 13 4 x 12 12 Vertex: 12 , 12

x 2 4 x 4 4 3

2 Âª x 2 7Âº Â¬ Â¼

x 2 7

Vertex: 2, 7

6 4

Axis of symmetry: x

3 4x x2

4 x2 x

â&#x2C6;&#x2019;2

x2 4x 3

4 r

2 1

x-intercepts: 2 r 6. f t

7, 0

Vertex: 1, 3

Axis of symmetry: x

6 5

â&#x2C6;&#x2019;1

x 1

â&#x2C6;&#x2019;2

2 t 1 3

â&#x2C6;&#x2019;3

t 1

2 t 1 3

Axis of symmetry: t

2 2 Âª t 1 1Âº 1 Â¬ Â¼

2 t 1

1 2 x 5x 4

8. f x

2 t 2 2t 1 1 1

x-intercepts: none

2t 2 4t 1

1 2

No real zeros

x 12

4 1 3

4 x

Axis of symmetry: x

3 2

2 1

6 2 6 Â· , 0 Â¸Â¸ 2 Â¹

â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;1

t 1

5 2

x2 5x 4

52 4 1 4

5 r

2 1

5 r 41 2

â&#x2C6;&#x2019;8

â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;6

NOT FOR SALE

Review Exercises ffor Chapter 2

9. (a) x x y y

2x 2 y

1000

14. f x

x 500 x

500 x x 2 x 2 500 x 62,500 62,500 x 250 62,500

The maximum area occurs at the vertex when x 250 and y 500 250 250. The dimensions with the maximum area are x 250 meters and y 250 meters.

The minimum cost occurs at the vertex of the parabola. 120 | 1091 units 2 0.055

About 1091 units should be produced each day to yield a minimum cost. x 4 , f x

11. y

3 4

16. f x

3 x 2 2

x7 8 x 2 8 x

The degree is odd and the leading coefficient is negative. The graph rises to the left and falls to the right. 17. g x

70,000 120 x 0.055 x 2

b Vertex: 2a

15. g x

The degree is even and the leading coefficient is positive. The graph rises to the left and rises to the right.

10. C

1 x3 2

The degree is odd and the leading coefficient is positive. The graph falls to the left and rises to the right.

500 x x 2

(b) A

2 x 2 5 x 12

The degree is even and the leading coefficient is negative. The graph falls to the left and falls to the right.

500 x

y A

13. f x

221

2 x3 4 x 2

(a) The degree is odd and the leading coefficient, 2, is positive. The graph falls to the left and rises to the right. (b) g x

2 x3 4 x 2

2 x 2 x 2

x 2 x 2

6 x4

2, 0

Zeros: x

(c)

5 4 3

â&#x20AC;&#x201C;3

g x

â&#x20AC;&#x201C;18

2 1 1

(d)

â&#x2C6;&#x2019; 4 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2

4 3

Transformation: Reflection in the x-axis and a vertical shift six units upward

(â&#x2C6;&#x2019; 2, 0) â&#x2C6;&#x2019;4 â&#x2C6;&#x2019; 3

x , f x

1 x5 2

12. y

(0, 0) â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1

x 3

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;4

8 6 4

â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;2

x 2

Transformation: Vertical shrink and a vertical shift three units upward

222

NOT FOR SALE

Chapter 2

18. h x

Polynomial ynomial and Rational Functions

x x3 x 2 5 x 3

20. f x

3x 2 x 4

(a) The degree is even and the leading coefficient, 1, is negative. The graph falls to the left and falls to the right.

(a) The degree is even and the leading coefficient is positive. The graph rises to the left and rises to the right.

(b) g x

3x 2 x 4

(b) Zeros: x

3x 2 x 4

(c)

x 3 x 2

â&#x20AC;&#x201C;4

â&#x20AC;&#x201C;1

100

â&#x20AC;&#x201C;18

â&#x20AC;&#x201C;8

(d) 0

â&#x20AC;&#x201C;4

x f x

â&#x20AC;&#x201C;2

h x

0, r

Zeros: x (c)

0, 1, 3

(â&#x2C6;&#x2019; 3, 0) 3

â&#x2C6;&#x2019;4

â&#x20AC;&#x201C;4

(1, 0) x

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;1

(0, 0)

(d)

â&#x2C6;&#x2019; 15

(0, 0)

â&#x2C6;&#x2019; 18 â&#x2C6;&#x2019; 21

(â&#x2C6;&#x2019; 3, 0(

( 3, 0(

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019; 3

â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1

21. (a) f x

3x3 x 2 3

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;4

19. f x

f x

x x 2 3

(b) Zero: x x f x

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C; 87

â&#x20AC;&#x201C; 25

â&#x20AC;&#x201C;1

(a) The degree is odd and the leading coefficient is negative. The graph rises to the left and falls to the right. (c)

â&#x20AC;&#x201C;3

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;6

(b) Zero: x | 0.900 22. (a) f x

x f x

x4 5x 1

â&#x20AC;&#x201C;3

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;1

â&#x20AC;&#x201C;5

There are zeros in the intervals > 1, 0@ and >1, 2@.

(d)

The zero is in the interval > 1, 0@.

(b) Zeros: x | 0.200, x | 1.772

3 2

(â&#x2C6;&#x2019;1, 0)

1 x

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019; 3 â&#x2C6;&#x2019; 2

6x 3 23. 5 x 3 30 x 2 3x 8 30 x 2 18 x

â&#x2C6;&#x2019;3

15 x 8

â&#x2C6;&#x2019;4

15 x 9 17 30 x 2 3 x 8 5x 3

6x 3

17 5x 3

NOT FOR SALE

Review Exercises ffor Chapter 2

26. f x

5x 4 24. x 2 5 x 1 5 x3 21x 2 25 x 4

3x3 8 x 2 20 x 16

(a) 4

5 x3 25 x 2 5 x 4 x 2 20 x 4

29 2

(b) â&#x20AC;&#x201C; 4

20 x 4 9 x3 14 x 2 3 x

(a) â&#x20AC;&#x201C;1

9 â&#x20AC;&#x201C; 20

14 11

â&#x20AC;&#x201C;3 3

0 0

â&#x20AC;&#x201C;11

â&#x20AC;&#x201C;3

20 Yes, x 3 4

20 Yes, x (c) 0

â&#x20AC;&#x201C;3

3 3

14 0

â&#x20AC;&#x201C;3 0

0 0

â&#x20AC;&#x201C;14

â&#x20AC;&#x201C;3

27. f x

0 is a zero of f. 9 20

14 29

â&#x20AC;&#x201C;3 15

0 12

â&#x20AC;&#x201C;16

â&#x20AC;&#x201C;4

4 is a zero of f. â&#x20AC;&#x201C;8

â&#x20AC;&#x201C; 20

â&#x20AC;&#x201C;12

â&#x20AC;&#x201C; 240

â&#x20AC;&#x201C; 20

â&#x20AC;&#x201C; 224

â&#x20AC;&#x201C;8

â&#x20AC;&#x201C; 20

â&#x20AC;&#x201C;4

â&#x20AC;&#x201C;16

â&#x20AC;&#x201C;6

â&#x20AC;&#x201C; 24

2 3

is a zero of f. â&#x20AC;&#x201C;8

â&#x20AC;&#x201C; 20

â&#x20AC;&#x201C;3

â&#x20AC;&#x201C;11

â&#x20AC;&#x201C;9

2 x3 11x 2 21x 90; Factor: x 6

(a) â&#x20AC;&#x201C;6

1 is not a zero of f.

No, x

9 0

No, x

(d) â&#x20AC;&#x201C;1

is a zero of f.

Yes, x (d) 1

3 4

Yes, x

2 3

1 is a zero of f.

4 is not a zero of f.

No, x (c)

â&#x20AC;&#x201C; 20

3 3

25. f x

(b)

5 5 x 4, x z r 2

â&#x20AC;&#x201C;8

Yes, x

0 5 x3 21x 2 25 x 4 x2 5x 1

3 3

4 x 2 20 x 4

223

â&#x20AC;&#x201C; 90

â&#x20AC;&#x201C;12

â&#x20AC;&#x201C;1

â&#x20AC;&#x201C;15

Yes, x 6 is a factor of f x . (b) 2 x 2 x 15

1 is not a zero of f.

2 x

5 x 3

The remaining factors are 2 x 5 and x 3 . (c) f x

2 x

5 x 3 x 6

(d) Zeros: x

52 , 3, 6

(e)

â&#x2C6;&#x2019;7

â&#x2C6;&#x2019; 100

224

NOT FOR SALE

Chapter 2

28. f x

Polynomial ynomial and Rational Functions

29. 8

x 4 4 x3 7 x 2 22 x 24

Factors: x 2 , x 3

(a) â&#x20AC;&#x201C;2

1 1

30. 5i i 2

â&#x20AC;&#x201C;4

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;10

â&#x20AC;&#x201C;24

â&#x20AC;&#x201C;6

100

â&#x20AC;&#x201C;9

â&#x20AC;&#x201C;12

â&#x20AC;&#x201C;3

â&#x20AC;&#x201C;4

8 10i

1 5i

31. 7 5i 4 2i

4 5i 2i

3 7i

Yes, x 2 and x 3 are both factors of f x . (b) x 2 3 x 4

1 x 4

The remaining factors are x 1 and x 4 . (c) f x

1 x 4 x 2 x 3

(d) Zeros: 2, 1, 3, 4 (e)

â&#x2C6;&#x2019;3

5 â&#x2C6;&#x2019; 10

32.

0 2 2i 2 2i

33. 7i 11 9i

77i 63i 2

34. 1 6i 5 2i

63 77i

36.

4 2 3i 2 1 i Â&#x2DC; Â&#x2DC; 2 3i 2 3i 1 i 1 i 8 12i 2 2i 4 9 1 1 8 12 i 1 i 13 13 Â§8 Â· Â§ 12 Â· 1Â¸ Â¨ i i Â¸ Â¨ Â© 13 Â¹ Â© 13 Â¹

4 2 2 3i 1 i

5 2i 30i 12i 2 5 28i 12 17 28i

35.

6 i 4 i

6 i 4 i Â&#x2DC; 4 i 4 i 24 10i i 2 16 1 23 10i 17 23 10 i 17 17

21 1 i 13 13

37. x 2 2 x 10

b r

b 4ac 2a 2

2 r 2r

2 1

4 1 10

2 2 r 6i 2 3i 1 r 3i

NOT FOR SALE

Review Exercises ffor Chapter 2

38. 4 x 2 4 x 7 x

b r

43. g x

b 2 4ac 2a

â&#x20AC;&#x201C;2

4 r

(4) 2 4(4)(7) 2(4)

4 r

1 r 2

â&#x20AC;&#x201C;7

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;36

â&#x20AC;&#x201C;9

g x

44. f x

6 i 2

39. Since g x

t 2 t 5 is a 5th degree polynomial

â&#x20AC;&#x201C;72

â&#x20AC;&#x201C;153

123

153

â&#x20AC;&#x201C;3

function, it has five zeros. 41. f x

Possible rational zeros: r1, r 2, r 3, r 4, r 5, r 6, r 10, r 12, r 15, r 20, r 30, r 60

â&#x20AC;&#x201C;2

â&#x20AC;&#x201C;30

1 1

x3 3 x 2 28 x 60

The zeros of f x are x 42. f x

x x

1 i

2 x 6 x 5

2, x

6, and x

f x

â&#x20AC;&#x201C;51

45. h x

8 2 4 1 17

2 1

8 r

4 2

4 r i.

3 x 3 x 4 i x 4 i

2 x5 4 x3 2 x 2 5

h x has three variations in sign, so h has either three or h x

0. Because 1 i is a zero, so is 1 i. â&#x20AC;&#x201C;6

4 4i

11 3i

7 4i

3 3i

6 0

7 4i

3 3i

4 4i

3 3i 0

â&#x20AC;&#x201C;24

one positive real zeros.

â&#x20AC;&#x201C;3

8 r

f x

2 x 2 x 30

The zeros of f x are 3, 3, 4 i, 4 i.

x 4 x3 11x 2 14 x 6

1 i

By the Quadratic Formula, the zeros of x 2 8 x 17 are x

4 x 4 11x3 14 x 2 6 x

One zero is x

x3 3 x 2 28 x 60

2, 3, 6.

2 x 3 x 6

x 2 x 8 is a 2nd degree polynomial

function, it has two zeros.

3 x 6 are

x 4 8 x3 8 x 2 72 x 153

40. Since h t

3, 6. The zeros of g x are x

4 r 4 6i 8

x3 7 x 2 36

The zeros of x 2 9 x 18

225

x ÂŞÂŹ x 1 i ÂşÂŞ ÂźÂŹ x 1 i ÂşÂź 4 x 3

2 x 4 x 2 x 5 5

2 x5 4 x3 2 x 2 5 h x has two variations in sign, so h has either two or no negative real zeros. 46. f x

(a)

4 x3 3 x 2 4 x 3

4 4

â&#x20AC;&#x201C;3

Because the last row has all positive entries, x is an upper bound.

x x 1 i x 1 i 4 x 3

Zeros: 0, 34 , 1 i, 1 i

226

Chapter 2

NOT FOR SALE

Polynomial ynomial and Rational Functions

47. Because the denominator is zero when x 10

x f x

0, the domain of f is all real numbers except x

10.5

10.1

10.01

10.001

o 10

303

3003

30,003

o f

10 m

9.999

9.99

9.9

9.5

29,997

2997

297

10.

As x approaches 10 from the left, f x increases without bound. As x approaches 10 from the right, f x decreases without bound.

48. Because the denominator is zero when x 2 10 x 24

4 and x

x x

f x

x f x

4 x 6

0, the domain of f is all real numbers except

3.5

3.9

3.99

3.999

o 4

2.667

6.4

38.095

398.010

3998.001

o f

4 m

4.001

4.01

4.1

4.5

f m

4002.001

402.010

42.105

10.67

As x approaches 4 from the left, f x increases without bound. As x approaches 4 from the right, f x decreases without bound. x f x

x f x

5.5

5.9

5.99

5.999

o 6

10.67

42.015

402.010

4002.001

o f

6 m

6.001

6.01

6.1

6.5

f m

3998.001

398.010

38.095

6.4

2.667

As x approaches 6 from the left, f x decreases without bound. As x approaches 6 from the right, f x increases without bound. 4 x

49. f x

(a) Domain: all real numbers x except x

(b) No intercepts (c) Vertical asymptote: x Horizontal asymptote: y (d)

0 0 y

f x

4 3

4 3 2 1 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019; 1

x 1

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;3

NOT FOR SALE

Review Exercises ffor Chapter 2

x 4 x 7

50. h x

(d)

(a) Domain: all real numbers x except x

(b) x-intercept: 4, 0

h x

2 3

5 8

4 7

1 2

2 5

1 4

4Â· Â¸ 7Â¹

1 2

227

8 6

Horizontal asymptote: y

)0, 47 )

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019; 2

(4, 0) x 2

10 12

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;8

x x 1

51. f x

6 x 2 11x 3 3x 2 x 3x 1 2 x 3

53. f x

(a) Domain: all real numbers x (b) Intercept: 0, 0

f x

2 5

(a) Domain: all real numbers x except x 1 x 3

1 2

2 5

2x 3 1 , x z x 3

x 3x 1

0 and

Horizontal asymptote: y

(d) 1

(0, 0)

x 1

f x

7 2

1 2

5 4

â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;2

52. y

2x2 x 4 2

(a) Domain: all real numbers x except x (b) Intercept: 0, 0

2 â&#x2C6;&#x2019;8 â&#x2C6;&#x2019; 6 â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;4

2, x

Horizontal asymptote: y (d)

x 4 3 ,0 2

( (

â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;8

50 21

8 3

18 5

2 3

y 6 4

(0, 0) â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;4

x 4

228

NOT FOR SALE

Chapter 2

Polynomial ynomial and Rational Functions

6x2 7 x 2 4x2 1 2 x 1 3x 2

54. f x

3x 2 1 , x z 2x 1 2

2 x 1 2 x 1

(a) Domain: all real numbers x except x

(d)

Horizontal asymptote: y 3

11 5

f x

8 3

f x

1 2 3 2

x 1

37 5

3 2 13 2

1 2 5 2

17 5

y 4

(0, 1)

0 2

2 x 1

(b) y-intercept: 0, 1

1 r 2

x 1

(a) Domain: all real numbers x except x

(b) y-intercept: 0, 2

(d)

x2 1 x 1

56. f x

2 3

1 3

â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;2

4 5

â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;2

528 p , 0 d p 100 100 p

57. C

(a)

â&#x2C6;&#x2019;1

( 23, 0(

4000

(0, â&#x2C6;&#x2019;2)

2 x3 x2 1

55. f x

2x 2 x 1

(a) Domain: all real numbers x (b) Intercept: 0, 0

100

f x

16 5

528 25

(b) When p

25, C

100 25

When p

50, C

100 50

When p

75, C

528 50

528 75

100 75

$176 million. $528million. $1584 million.

y 3 2 1

(0, 0) â&#x2C6;&#x2019;3

â&#x2C6;&#x2019;2

â&#x2C6;&#x2019;1

x 1

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;3

NOT FOR SALE

Review Exercises ffor Chapter 2

58. (a)

61.

2 in. y 2 in.

2 in. 2 in.

2 3 d x 1 x 1 2 x 1 3 x 1

d 0 x 1 x 1

2 x 2 3x 3 d 0 x 1 x 1

(b) The area of print is x 4 y 4 , which is 30 square inches.

4 y 4

Test: Is

62.

2 x 2 x 7

x 4

1 3 x 2 0 1 4

Test: Is 4 x 1 3 x 2 0? By testing an x-value in each test interval in the Â§ 2 1Âˇ inequality, you see that the solution set is Â¨ , Â¸. ÂŠ 3 4Âš

x x 4 x 4 t 0 0, x

d 0?

0, x

4, x

4 x 5

2000 d

1000 1 3t

5 t 1000 1 3t

5 t 2000 5 t d 1000 1 3t

10,000 2000t d 1000 3000t 1000t d 9000 t t 9 days 64. An asymptote of a graph is a line to which the graph becomes arbitrarily close as x increases or decreases without bound.

x3 16 x t 0

Key numbers: x

63.

2Âˇ Â§ 2 1Âˇ Â§1 Âˇ Â§ Test intervals: Â¨ f, Â¸, Â¨ , Â¸, Â¨ , f Â¸ 3Âš ÂŠ 3 4Âš ÂŠ 4 Âš ÂŠ

60.

1 x 1

0? x By testing an x-value in each test interval in the inequality, you see that the solution set is f, 0 Â&#x2030; 4, 5 .

12 x 2 5 x 2 0

2 ,x 3

x 2 9 x 20 0 x x 4 x 5 0 x

Test: Is

12 x 5 x 2

Key numbers: x

Test intervals: f, 0 , 0, 4 , 4, 5 , 5, f

4 x

5, x

x 5

Key numbers: x

d 0

By testing an x-value in each test interval in the inequality, you see that the solution set is [ 5, 1) Â&#x2030; 1, f .

ÂŞ 2 2 x 7 Âş xÂŤ Âť ÂŹ x 4 Âź

1 x 1

Test intervals: f, 5 , 5, 1 , 1, 1 , 1, f

x 4 4 x 14 x 4 2 2 x 7

x 4

x 5

Key numbers: x

30 x 4 30 4 x 4 30 4 x 4

Total area

y 4

229

Test intervals: f, 4 , 4, 0 , 0, 4 , 4, f

65. False. A fourth-degree polynomial can have at most four zeros, and complex zeros occur in conjugate pairs.

Test: Is x x 4 x 4 t 0?

66. False.

By testing an x-value in each test interval in the inequality, you see that the solution set is > 4, 0@ Â&#x2030; [4, f).

The domain of f x

1 x2 1

is the set of all real numbers x.

230

NOT FOR SALE

Chapter 2

Polynomial ynomial and Rational Functions

Problem Solving for Chapter 2 1. f x

ax3 bx 2 cx d

ax 2 ak b x ak 2 bk c

x k ax3 bx 2

ax akx 3

ak ak

b x 2 cx b x 2 ak 2 bk x

ak 2 ak 2

bk c x d bk c x ak 3 bk 2 ck

ak 3 bk 2 So, f x

f x

2. (a)

ax 3 bx 2 cx d

ck d

k ÂªÂ¬ax 2 ak b x ak 2 bx c ÂºÂ¼ ak 3 bk 2 ck d and

ak 3 bk 2 ck d . Because the remainder is r

y3 y 2

y 1

150

252

392

576

810

1100

ak 3 bk 2 ck d , f k

(iv) 2 x3 5 x 2

2500; a

(ii)

(v) 7 x3 6 x 2 a

(vi) 10 x3 3x 2

x3 2 x 2

288; a

1 3 1 2 x 2 x

8 8 3

(iii) 3x x 3

1, b

3, b

9 90

3 x

810 Â&#x; 3 x

3 x

2 Â&#x;

a2 b3

1 8

3 Â&#x; x

a2 b3

a2 1Â&#x; 3 b

9 Â&#x; x

4 Â&#x; x

49 216 49 1728

216

392 Â&#x;

7x 6

7 Â&#x; x

297; 3Â&#x;

10, b

a2 b3

100 27

100 10 x3 100 3x 2

2x 5

x 36 Â&#x; 2

90; a

1 288

9 3 x3 9 x 2 3

80 Â&#x;

6 Â&#x;

7, b

4 125

1728;

252 Â&#x; x

a2 b3

4 2500

125

49 49 7 x3 216 6 x 2

216

x3 x 2

5Â&#x;

2, b

4 4 2 x3 125 5 x 2

125 3

(b) (i)

100 297

1100 Â&#x;

10 x 3

10 Â&#x; x

NOT FOR SALE

Problem Solving ffor Chapter 2

x 2 x 3

l Â&#x2DC;wÂ&#x2DC;h

3. V

x 2 x 3

x3 3 x 2 20

1 1

2 x 2 5 x 10

5 r

2 or x

4 1 2 1

(b) Slope

4.41 4 2.1 2

f 2 h f 2

(d) Slope

q x

r x

5. (a) y

f x

q x

x 1

f 1

x 1

4, 0 :

1, 0 :

a 4 b 4 4

16a 4b 4

4a b 1

a 1 b 1 4

a b

a 1 4a

1 3a

1 4 1

5 4.1

4 4a b 1

k q x r

One possibility:

f x

2 x 2 5

x3 2 x 2 5

1 4a

(b) Cubic, passes through 3, 1 , falls to the right

4 1 4 0.1

(a) Cubic, passes through 2, 5 , rises to the right

7. f x

4 1

h z 0

(f ) Letting h get closer and closer to 0, the slope approaches 4. So, the slope at 2, 4 is 4.

a 0 b 0 c

4 h,

The results are the same as in (a)â&#x20AC;&#x201C;(c). .

Slope

ax 2 bx c

0, 4 :

(e)

The statement should be corrected to read f 1

since

h 4

4h h 2 h 4 h, h z 0

d x q x r x , we have

d x

h 2 h

The dimensions of the mold are 2 inches u 2 inches u 5 inches.

f x

4.1

Slope of tangent line is less than 4.1.

x+3 x

15i

4. False. Since f x

Slope of tangent line is greater than 3.

Choosing the real positive value for x we have: x 2 and x 3 5.

d x

Slope of tangent line is less than 5.

Possible rational zeros: r1, r 2, r 4, r 5, r10, r 20 2

9 4 3 2

6. (a) Slope

231

One possibility: f x

x 3 x 2 1 x3 3x 2 1

x2 5x 4

(b) Enter the data points 0, 4 , 1, 0 , 2, 2 , 4, 0 ,

6, 10

and use the regression feature to obtain

x 5 x 4. 2

232

Chapter 2

8. (a) zm

Polynomial ynomial and Rational Functions

1 z

1 1 i 1 i 2 (b) zm

10. f x

1 2 8i 1 2 8i Â&#x2DC; 2 8i 2 8i 2 8i 68

1 z 1 3 i Â&#x2DC; 3 i 3 i 3 1 i 10 10

1 3 i 3 i 10

1 z

9. a bi a bi

1 2 i 34 17

a 2 abi abi b 2i 2

a 2 b2

Since a and b are real numbers, a 2 b 2 is also a real number.

ax b cx d

Vertical asymptote: x Horizontal asymptote: y

d c a c

(i) a ! 0, b 0, c ! 0, d 0 Both the vertical asymptote and the horizontal asymptote are positive. Matches graph (d). (ii) a ! 0, b ! 0, c 0, d 0 Both the vertical asymptote and the horizontal asymptote are negative. Matches graph (b). (iii) a 0, b ! 0, c ! 0, d 0 The vertical asymptote is positive and the horizontal asymptote is negative. Matches graph (a). (iv) a ! 0, b 0, c ! 0, d ! 0 The vertical asymptote is negative and the horizontal asymptote is positive. Matches graph (c). 11. f x

(a) b z 0 Â&#x; x

b is a vertical asymptote.

a causes a vertical stretch if a ! 1 and a vertical shrink if 0 a 1. For a ! 1, the graph becomes wider as a increases. When a is negative, the graph is reflected about the x-axis. (b) a z 0. Varying the value of b varies the vertical asymptote of the graph of f. For b ! 0, the graph is translated to the right. For b 0, the graph is reflected in the x-axis and is translated to the left.

NOT FOR SALE

Problem Solving ffor Chapter 2

12. (a)

Age, x

(c)

Near point, y

3.0

4.7

233

Age, x Near point, y Quadratic Model Rational Model 16

3.0

3.66

3.05

4.7

2.32

4.63

9.8

11.83

7.58

19.7

19.97

11.11

39.4

38.54

50.00

y | 0.031x 2 1.59 x 21.0 (b)

1 | 0.007 x 0.44 y 1 y | 0.007 x 0.44 50

The models are fairly good fits to the data. The quadratic model seems to be a better fit for older ages and the rational model a better fit for younger ages. (d) For x 25, the quadratic model yields y | 0.625 inch and the rational model yields y | 3.774 inches. (e) The reciprocal model cannot be used to predict the near point for a person who is 70 years old because it results in a negative value y | 20 . The quadratic model yields y | 63.37 inches.

13. Because complex zeros always occur in conjugate pairs, and a cubic equation has three zeros and not four, a cubic equation with real coefficients can not have two real zeros and one complex zero.

234

Chapter 2

NOT FOR SALE

Polynomial ynomial and Rational Functions

Practice Test for Chapter 2 1. Sketch the graph of f x

x 2 6 x 5 and identify the vertex and the intercepts.

2. Find the number of units x that produce a minimum cost C if

0.01x 2 90 x 15,000.

3. Find the quadratic function that has a maximum at 1, 7 and passes through the point 2, 5 . 4. Find two quadratic functions that have x-intercepts 2, 0 and

43 , 0 .

5. Use the leading coefficient test to determine the right and left end behavior of the graph of the polynomial function f x

3 x 5 2 x3 17. 6. Find all the real zeros of f x

x 5 5 x 3 4 x.

7. Find a polynomial function with 0, 3, and 2 as zeros. 8. Sketch f x

x 3 12 x.

9. Divide 3x 4 7 x 2 2 x 10 by x 3 using long division. 10. Divide x 3 11 by x 2 2 x 1. 11. Use synthetic division to divide 3x 5 13x 4 12 x 1 by x 5. 12. Use synthetic division to find f 6 given f x

7 x 3 40 x 2 12 x 15.

13. Find the real zeros of f x

x3 19 x 30.

14. Find the real zeros of f x

x 4 x 3 8 x 2 9 x 9.

15. List all possible rational zeros of the function f x

16. Find the rational zeros of the polynomial f x

17. Write f x

6 x 3 5 x 2 4 x 15.

20 2 x 3

10 . 3

x 4 x 3 5 x 10 as a product of linear factors.

18. Find a polynomial with real coefficients that has 2, 3 i, and 3 2i as zeros. 19. Use synthetic division to show that 3i is a zero of f x

20. Sketch the graph of f x

x 3 4 x 2 9 x 36.

x 1 and label all intercepts and asymptotes. 2x

21. Find all the asymptotes of f x

8x2 9 . x2 1

22. Find all the asymptotes of f x

4 x2 2 x 7 . x 1

Precalculus with Limits 3rd Edition Larson Solutions Manual Full Download: http://alibabadownload.com/product/precalculus-with-limits-3rd-edition-larson-solutions-manual/

NOT FOR SALE

Practice Test ffor Chapter 2

23. Given z1

4 3i and z2

235

2 i, find the following:

(a) z1 z2 (b) z1 z2 (c) z1 z2 24. Solve the inequality: x 2 49 d 0 25. Solve the inequality:

x 3 t 0 x 7

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