College Algebra Helper Preview by Mark Long

The College Algebra Helper Dr. Hossein Pezeshki

£ x

f £ ( g ( x)) 2

4 1 = + 2 2 x = ( a + b) + 1 y x b2

= y

P( x) =

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y = 3x -

f -1( f ( y ))

The College Algebra Helper

Dr. Hossein Pezeshki Texas State Technical College Harlingen

ÂŠ 2006 Hossein Pezeshki ISBN 0-9786773-2-3 All rights reserved, including the right to reproduce this book or any portion thereof in any form. Requests for such permissions should be addressed to: TSTC Publishing Texas State Technical College Waco 3801 Campus Drive Waco, TX 76705 http://publishing.tstc.edu/ Publisher: Mark Long Graphics specialist: Katherine Wilson Interior page layout: Grace Arsiaga Editor: Christopher Wilson Printing production: Bill Evridge Cover design: Philip Dunnells Editorial interns: Meredith Recer, Kate McClendon, and Cheyenne Feeney Graphic interns: Jaclyn Henderson, Jerri Tudor, JosĂŠ Miguel Reyes, and Marilee Schwertner

Manufactured in the United States of America First edition

To my wife Mary and my children David and Leila

Table of Contents

Chapter 1: Graphs and Functions

Section 1.1: Functions and One-to-One Functions..................................................................................1 Section 1.2: More on Functions................................................................................................................11 Section 1.3: Combining Functions...........................................................................................................17 Section 1.4: Inverse of a Function............................................................................................................23

Chapter 2: Functions, Equations and Inequalities

Section 2.1: Complex Numbers................................................................................................................29 Section 2.2: Quadratic Equations.............................................................................................................41 Section 2.3: Solving Linear Inequalities..................................................................................................53

Chapter 3: Polynomial and Rational Functions

Section 3.1: Synthetic Division — Remainder and Factor Theorem...................................................65 Section 3.2: Real Zeros of Polynomials — Part A.................................................................................75 Section 3.3: Real Zeros of Polynomials — Part B.................................................................................87 Section 3.4: Graphs of Polynomial Functions.......................................................................................93 Section 3.5: Rational Functions and Their Graphs................................................................................99

Chapter 4: Exponential and Logarithmic Functions

Section 4.1: Exponential Functions........................................................................................................109 Section 4.2: Exponential Functions with Base e...................................................................................119 Section 4.3: Graphs of Logarithmic Functions.....................................................................................127 Section 4.4: Properties of Logarithmic Functions................................................................................137 Section 4.5: Solving Exponential Equations.........................................................................................153 Section 4.6: Solving Logarithmic Equations........................................................................................159

Chapter 5: Systems of Equations and Matrices

Section 5.1: Solving Linear Systems Using Matrices...........................................................................167 Section 5.2: Operations with Matrices – Part A (Addition & Subtraction with Matrices).............179 Section 5.3: Operations with Matrices – Part B (Multiplication with Matrices)..............................189 Section 5.4: Inverse of a Matrix..............................................................................................................195 Section 5.5: Solution of Linear Systems in Two and Three Variables Using Determinants..........203 Section 5.6: Partial Fractions...................................................................................................................217

Chapter 6: Sequences and Series

Section 6.1: Sequences............................................................................................................................ 227 Section 6.2: Sums and Series...................................................................................................................233 Section 6.3: Arithmetic Sequences.........................................................................................................241 Section 6.4: Arithmetic Series and Their Sums....................................................................................251 Section 6.5: Geometric Sequences..........................................................................................................257 Section 6.6: Geometric Series and Their Sums.....................................................................................265

Author Bio...................................................................................................................271 About TSTC Publishing.............................................................................................273

Functions and One-to-One Functions

Section 1.1: Functions and One-to-One Functions After completing this section you will be able to: • Define a function • Define a one-to-one function

Definition of a Function A function is a relationship between two variables x and y, such that for every value of x there is a corresponding value of y. Then we say y is a function of x and we write it as f(x) = y. The x variable is called an independent variable or input value or domain. The y variable is called a dependent variable, output value, or range. Consider the following relations given as a set of ordered pairs: f = {(-1, 0), (0, 1), (1, 2), (2, 3), (-2,-1)} g = {(0, 0), (1, 1), (-1, 1), (2, 4), (-2, 4)} h = {(0, 0), (1, 1), (1, -1), (4, 2), (4, -2)} Let’s write these sets of ordered pairs in a table as below: f Domain (x-value)

Range (y-value)

-1 0 1 2 -2

0 1 2 3 -1

f = {(-1, 0), (0, 1), (1, 2), (2, 3), (-2,-1)} Since no number of the domain column is repeated, the relation is called a function. Also since no number in the range column is repeated, the relation is a one-to-one function.

Functions and One-to-One Functions

g Domain (x-value)

Range (y-value)

0 1 -1 2 -2

0 1 1 4 4

g = {(0, 0), (1, 1), (-1, 1), (2, 4), (-2, 4)} Since no number of the domain column is repeated, the relation is called a function. But since some number of the range is repeated, then the relation is not a one-to-one function. h Domain (x-value)

Range (y-value)

0 1 1 4 4

0 1 -1 2 -2

h = {(0, 0), (1, 1), (1, -1), (4, 2), (4, -2)} Since some numbers of the domain are repeated the relation is not a function. If a relation is not a function it is not one-to-one either.

Graph of a Function EXAMPLE 1 Graph f ( x ) = 1 â&#x2C6;&#x2019; x and give the domain, range, and zeros. Solution STEP 1 Let the term inside the absolute value be equal to zero and solve for x. 1 - x = 0, then x = 1, hence 1 would be the starting value for the domain of the function.

Functions and One-to-One Functions

STEP 2 Construct a table of x and y values as follows: x

-1

STEP 3 Graph.

STEP 4 Find the domain and range. D: (-∞, ∞) R: [0, ∞)

EXAMPLE 2 Graph f ( x ) =

16 − x 2 and give the domain, range, and zeros.

Solution STEP 1 Let the terms under the square root be equal to zero and solve for x. 16 − x 2 = 0 ⇒ 16 = x 2 ⇒ x = ±4 , Hence -4 and 4 would be the starting values for the domain of the function. STEP 2 Construct a table of x and y values as follows: x

-5

-4

Functions and One-to-One Functions

STEP 3 Graph.

STEP 4 Find the domain and range. D: [-4, 4] R: [0, 4]

EXAMPLE 3 Graph f ( x ) =

x and give the domain, range, and zeros.

Solution STEP 1 Let the term under the square root be equal to zero and solve for x = 0, hence 0 would be the starting value for the domain of the function. STEP 2 Construct a table of x and y values as follows: x

-1

STEP 3 Graph. y

Functions and One-to-One Functions

STEP 4 Find the domain and range. D: [0, â&#x2C6;&#x17E;) R: [0, â&#x2C6;&#x17E;)

Vertical Line Test A relation is a function if a vertical line crosses its graph at only one point.

Horizontal Line Test A relation is a one to one function if a horizontal line crosses its graph at only one point.

EXAMPLE 4 Apply the vertical and horizontal line tests to determine which is a graph of a function, and which is a one-to-one function. Example-4a. y

Example-4b. y

function and one-to-one function

function, but not one-to-one function

Example-4c. y

Example-4d. y

not a function or one-to-one function

function but not one-to-one function

Functions and One-to-One Functions

EXAMPLE 5 Graph the function y = x2 and then determine whether the graph is a graph of a function and/or a one-to-one function. Solution STEP 1 Construct a table of x and y values as follows. Let 0 be the magic number for the domain of the function. x

-2

-1

STEP 2 Graph the function. y

STEP 3 The vertical line test shows that this is a graph of a function, and the horizontal line test shows that this is not a graph of a one-to-one function.

EXAMPLE 6 Graph the function y = x and then determine whether the graph is the graph of a function and/or a one-to-one function. Solution STEP 1 Construct a table of x and y values as follows: x

-1

Functions and One-to-One Functions

STEP 2 Graph the function. y

STEP 3 The vertical and horizontal line tests show that this is a graph of a function and also a one-to-one function.

Functions and One-to-One Functions

Section 1.1 Exercises: Functions and One-to-One Functions Name___________________________ Date____________________________ 1. Give an example of a function and one to one function in a set of ordered pairs.

2. Use the vertical line and the horizontal line tests to determine whether the graph is a graph of a function or one-to-one function. a. b. y y

c. y

d. y

Functions and One-to-One Functions

3. Graph the following functions, then determine whether each graph is a graph of a function or one-to-one function. a. y = x2 + 1

b. y = x + 2

4.Give the domain of each function. Do not graph. a. y = x

b. y = x 2 − 1

c. y =

x+1 x2 – 4

5. Graph the given functions. Find the domain and range. 2 a. y = 16 − x

b. yy == 4 − x

c. y = x2 – 1

d. y = x2 + 1

More On Functions

Section 1.2: More On Functions After completing this section you will be able to: • Graph a given function and determine the intervals on which the function is increasing, decreasing or constant • Graph a given function and determine the coordinates of relative maximum and relative minimum • Define and graph a piecewise function

Definition of an Increasing, Decreasing, and Constant Function A function is said to be: • Increasing when the graph of a function rises from left to right on a given interval • Decreasing when the graph of a function drops from left to right on a given interval • Constant when the graph of a function stays the same from left to right When finding the intervals on which a function is increasing, decreasing or constant, use the values on the x-axis.

Intervals EXAMPLE 1 Determine the intervals on which the graph below of f(x) is: A) increasing B) decreasing or C) constant. y 5

-3

Solution A f(x) is increasing on [-3, 0).

More On Functions

Solution B f(x) is decreasing on (4, ∞). Solution C f(x) is constant on (0, 4).

EXAMPLE 2 Determine the intervals on which the graph below of g(x) is: A) increasing B) decreasing or C) constant. y

Solution A g(x) is increasing on [-3, 0). Solution B g(x) is decreasing on (0, 3]. Solution C g(x) has no interval on which it is constant.

Piecewise Functions A function that has more than one part is called a piecewise function.

EXAMPLE 3 Graph the piecewise function, find A) domain B) range C) interval on which f(x), increasing, decreasing, or constant. x≥0  9  2 f(x) =  9 − x -3 ≤ x < 0 -9 − 2x x < -3  We need to construct tables for each part of the function based on given requirements.

More On Functions

STEP 1 For f(x) = 9, x ≥ 0, the function doesn’t need a table. STEP 2 For f(x) = 9 – x2, -3 ≤ x < 0, the table is as follows: x

-3

-2

-1

STEP 3 For f(x) = -9 – 2x, x < -3, the table is as follows: x

-5

-4

-3

-1

-3

STEP 4

y 9

-3 -3 Solution A Domain (-∞, ∞) Solution B Range (-3, 9] Solution C Increasing (-3, 0) Decreasing (-∞, -3) Constant (Ø, ∞)

More On Functions

Relative Maximum The highest point on the graph of a function in some open interval (hill of the graph).

Relative Minimum The lowest point on the graph of a function in some open interval (valley of the graph).

EXAMPLE 4 Given the graph of f(x), find the coordinates of relative maximum and relative minimum. y 4 x

3 -3

Solution Relative maximum at (0, 4) Relative minimum at (3, -3)

EXAMPLE 5 Given the graph of f(x), find the coordinates of relative maximum and relative minimum. y

x Solution Relative maximum at (0, 5) Relative minimum at (-2, 0) and (2, 0)

More On Functions

Section 1.2 Exercises: More on Functions Name________________ Date_________________ 1. Use the following graphs to A) determine the interval on which the function is increasing, decreasing and constant B) find domain and range. a. b. y y

c. y

d. y

x x

More On Functions

2. Use the following graphs to determine the coordinates of relative maximum and relative minimum. a. b. y y 4 x

-3

3. Graph the piecewise function. Find A) domain and range B) determine the interval on which f(x) increasing, decreasing, or constant. f (x) =

{

x â&#x2030;¤ 0

x > 0

Combining Functions

Section 1.3: Combining Functions After completing this section you will be able to: • Add, subtract, multiply and divide functions and find their domains • Define the composition of functions

Adding, Subtracting, Multiplying and Dividing Functions For functions f(x) and g(x): (f + g)(x) = f(x) + g(x) (f – g)(x) = f(x) – g(x) (f • g)(x) = f(x) • g(x) f  f (x)   (x) = g( x ) g For example, given the graph of f and g shown below, find A) (f + g)(0) and B) (f + g)(-4). g(x)

f(x)

Solution A f(0) + g(0) = 4 + 4 = 8 Solution B f(-4) + g(-4) = 0 + 0 = 0

Combining Functions

EXAMPLE 1 Let f ( x ) = x 2 − 2 and g ( x ) = 5 +

x and find A) f(2) B) f(-2) C) g(-4) D) g(1) 2

Solution A f (2) =

22 − 2 =

Solution B f (-2) =

(-2) 2 − 2 =

Solution C g(-4) = 5 + Solution D g(1) = 5 +

-4 = 5 − 2 = 3 2 1 10 + 1 11 = = 2 2 2

EXAMPLE 2 Let f(x) = 5x + 2 and find A) f(k) B) f(k + 1) C) f(2p) Solution A f(k) = 5(k) + 2 = 5k + 2 Solution B f(k + 1) = 5(k + 1) + 2 = 5k + 7 Solution C f(2p) = 5(2p) + 2 = 10p + 2

EXAMPLE 3 Let f(x) = 2x2 – 3 and g(x) = x – 3 and find A) f(g(4)) B) g(f(-2)) Solution A Evaluate g(4) first g(4) = 4 – 3 = 1, then evaluate f(1) = 2(1)2 – 3 = -1 Solution B Evaluate f(-2) first f(-2) = 2(-2)2 – 3 = 5, then evaluate g(5) = 5 – 3 = 2

Combining Functions

EXAMPLE 4 Let f(x) = x2 + 2 and g(x) = x – 1 and find A) f + g B) f – g C) f • g D)

f and their domains. g

Solution A (f +g)(x) = (x2 + 2) + (x – 1) = x2 + x + 1 Domain all real numbers. Solution B f(x) - g(x) = (f – g)(x) = (x2 + 2) – (x – 1) = x2 + 2 – x + 1 = x2 – x + 3 Domain all real numbers. Solution C f(x) • g(x) = (f • g)(x) = (x2 + 2)(x – 1) = x3 – x2 + 2x – 2 Domain all real numbers. Solution D f  f (x) x2 + 2 =   (x) = g( x ) x −1 g Domain all real numbers except x = 1.

Composition of Functions We define the combinations of two functions such as f(x) and g(x) by writing (f ◦ g)(x) and (g ◦ f)(x). We call (f ◦ g)(x) as f(g(x)) the (f composite g) and (g ◦ f)(x) by g(f(x)) the (g composite f).

EXAMPLE 5 Let f(x) = x2 + 2 and g(x) = x – 1 and find A) f(g(x)) B) g(f(x)) Solution A f(g(x)) = f(x – 1) = (x – 1)2 + 2 = x2 – 2x + 1 + 2 = x2 – 2x + 3 Solution B g(f(x)) = g(x2 + 2) = (x2 + 2) – 1 = x2 + 2 – 1 = x2 + 1

Combining Functions

EXAMPLE 6 Let f(x) = x2 + x + 1 and g(x) = x + 1 and find A) f(g(x)) B) g(f(x)) Solution A f(g(x)) = (x + 1)2 + (x + 1) + 1 = x2 + 2x + 1 + x + 1 + 1 = x2 + 3x + 3 Solution B g(f(x)) = (x2 + x + 1) + 1 = x2 + x + 1 + 1 = x2 + x + 2

Combining Functions

Section 1.3 Exercises: Combining Functions Name_______________________ Date________________________ 1. Let f(x) = 5x – 10, g(x) = 7 – 3x. Find: a. f(-5)

b. g(0)

c. 2f(6) + 3g(-1)

d. f (-6) g (-2)

e. f(x + 1)

f. g(x + 1)

Combining Functions

2. Let f(x) = x2 + x – 1 and g(x) = x – 1. Find the indicated functions and their domains. a. f(g(x))

b. g(f(x))

c. (f + g)(x)

d. (f – g)(x)

e. (f • g)(x)

f. f ( x ) g( x )

Inverse of a Function

Section 1.4: Inverse of a Function After completing this section you will be able to: • Define an inverse of a one-to-one function • Write the inverse of a one-to-one function in form of a set of an ordered pairs • Write the formula for the inverse of a one-to-one function in form of an equation • Show that f(f –1(x)) = f –1(f (x)) = x • Graph a function and its inverse on the same rectangular coordinate system

Definition of an Inverse If y is a function of x, and for every value of x, there is a corresponding value of y (one to one function), then the inverse of one to one function is given by interchanging the value of x and y. Then we write the inverse of y = f(x) as y = f –1(x).

Rule for Writing the Inverse of a One-To-One Function in Set Form Given that f = {(0, 1), (1, 3), (-1,-1)} is a one-to-one function, then its inverse is given by f –1 = {(1, 0), (3, 1), (-1, -1)}. If f(x) is a one-to-one function then we say f –1(f(x)) = x and f(f –1(x)) = x.

Rule for Writing the Formula for the Inverse Function To find the inverse formula of a one-to-one function in form of an equation: 1. Let f(x) = y. 2. Interchange the x and y variables. 3. Solve for y. 4. Let y = f –1(x).

Inverse of a Function

EXAMPLE 1 Given f(x) = 4x – 1, find f –1(x). Solution STEP 1 f(x) = 4x – 1 STEP 2 y = 4x – 1 STEP 3 x = 4y – 1 STEP 4 x + 1 = y 4 STEP 5 x + 1 = f 4

−1

(x)

EXAMPLE 2 Show that f(x) = 4x – 1 and f(x) =

x+1 are inverses of each other. 4

Solution STEP 1 Let x = 2, then: STEP 2 Show that f –1(f(2)) = 2 STEP 3 Evaluate f(2) = 4(2) – 1 = 8 – 1 = 7 STEP 4 f −1 (f (2)) = f −1 (7) =

7 + 1 = 2 4

STEP 5 Hence, they are inverses of each other.

Inverse of a Function

EXAMPLE 3 Given f(x) = 3x – 5, find A) f –1(x) and B) f(f –1(-2)). Solution A STEP 1 y = 3x – 5 STEP 2 x = 3y – 5 STEP 3 x + 5 = 3y STEP 4 x + 5 = y 3 STEP 5 f −1 ( x ) =

x + 5 3

Solution B f(f -1(-2)) STEP 1 -2 + 5 f −1 (-2) = = 3 STEP 2 3 = 1 3 STEP 3 f(x)= 3x – 5 STEP 4 f(1) = 3(1) – 5 STEP 5 f(1) = 3 – 5 STEP 6 f(1) = -2 So, f(x) and f - 1 (x) are the inverses of each other.

Inverse of a Function

EXAMPLE 4 Graph y = 4x – 1 and its inverse. STEP 1 Construct a table of x and y values for f(x) and f –1(x). f(x) =>

-2

-1

-9

-5

-1

This table shows that the given equation is a one to one function. Hence, to construct a table for its inverse we interchange the values of x and y in above table STEP 2 f –1(x) =>

-9

-5

-1

-2

-1

STEP 3 Graph the function and its inverse on the same rectangular coordinate system. y f(x) x f -1(x)

Inverse of a Function

Section 1.4 Exercises: Inverse of a Function Name________________________ Date_________________________ 1. List rules for writing the formula for the inverse of a function.

2. Given that f (x) = 3x – 1. Find: a. f –1(x)

b. (f –1(3))

c. f –1(f (4))

3. Graph the function and its inverse. a. y = 2x – 1

4. Given f(x) =

b. y = 4x + 1

x+2 find: 4

a. f –1(x)

b. f –1(2)

c. f –1(f (2))

About the Author A native of Tehran, Iran, Dr. Hossein Pezeshki is chair of the mathematics department at Texas State Technical College Harlingen. A long-time instructor at TSTC, he also teaches pre-calculus and AP calculus at nearby Rio Hondo High School. He feels that a large part of his role as a teacher is to motivate his students to stay in school and emphasize the power of education to them. “To me,” he says, “there is no treasure better than an education.” But, as he also tells his students, “It takes time, it takes effort, and it takes desire if you want an education.”

TSTC Publishing Established in 2004, TSTC Publishing is a provider of high-end technical instructional materials and related information to institutions of higher education and private industry. â&#x20AC;&#x153;High endâ&#x20AC;? refers simultaneously to the information delivered, the various delivery formats of that information, and the marketing of materials produced. More information about the products and services offered by TSTC Publishing may be found at its Web site: http://publishing.tstc.edu/.

The First-Semester College Algebra Student’s Best Friend For the student (or teacher) who doesn’t want or need a 1,200 page textbook for first-semester algebra, The College Algebra Helper provides a focused, yet thorough, overview of quadratics; polynomial, rational, logarithmic, and exponential functions; systems of equations; progressions; sequences and series; and matrices and determinants.

Included in The College Algebra Helper:

• Clear, easy-to-understand language;

• Numerous sample problems done step-by-step; and

• Carefully tailored exercises at the end of each section. Dr. Hossein Pezeshki is chair of the math department at Texas State Technical College Harlingen. A dedicated and long-time educator, he feels that a large part of his role as a teacher is to motivate his students and emphasize the power of a good education. More information about this book as well as other books from TSTC Publishing may be found at: http://publishing.tstc.edu/

RVSD 08-08 ISBN 0-9786773-2-3

9 780978 677329

1 0 0 0 0