Chapter 1 Functions Exam 1.If y = x2 + 8x - 33, the values of x for which y = 0 are Ans: x = 3 and x = -11 x6 x , find g(x + 3). x+9 Ans: x + 3
g ( x)
2. If
3. Find the natural domain and range for f ( x) 3x 9 . 3, Ans: Domain: x 3 or 0, Range: y 0 or 4.Find the natural domain and range for 8x 3 f ( x) 5x 8 . 8 8 8 x , , 5 5 5 or Ans: Domain: y
Range:
8 5 or
8 8 , , 5 5
5.Find the natural domain for
f ( x) Ans: x 2 2, or
4 x2 .
6..Find the natural domain of
x3 x 1 . x 1
f ( x)
Ans: x 3 or ,3 1, or
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2 7. Find the natural domain for h( x) 70 9 x x . Ans: [-14, 5]
8. Find the natural domain for 6x f ( x) x 11 . Ans: x 11 or x 0 or
, 11 0,
9. Find f(8) if f(x) = x2 - 5x + 11. Ans: 35
10. What is the natural domain of the function –7 f ( x) 6 2 x 1 ? Ans: x 1 x4 3x g ( x) f 3 , write an expression for g(x) and find its range x 4 and 11. If and domain. 3x 12 g ( x) x 16 Ans:
f ( x)
Domain: Range:
x 16 y3
or or
, 16 16, ,3 3,
x2 4 x 1 f ( x) 3x 12. Let and g(x) = x - 2, find (f + g)(x). 2 4 x 10 x 1 ( f g )( x) 3x Ans:
3x 2 5 x 1 2x 13. Let and g(x) = x - 5, find (f - g)(x). 2 x 5x 1 ( f g )( x) 2x Ans: f ( x)
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f 2 x 2 4 x 48 x 7 x 14. If and g(x) = x - 6, find g . 2x 8 Ans: 7 x for x 6 f ( x)
2x 8 x 2 and g ( x) 7 x , find f g ( x) . 15. Let 2 7x Ans: 8 7 x f ( x)
16. Let g(x) = 5x + 8 and h(x) = 3x - 7, find Ans: 15t + 48 domain:
g h t 5
all real numbers
and its domain or
17. Let f ( x) x 8 and g ( x) 3 x , find g f ( x) and its domain. 4 Ans: 3 x 8 Domain:
x8
or
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8,
,
2 18. Express h( x) x 1 as the composition of two functions such that h(x) = f g(x). Ans: f ( x) x
g ( x) x 2 1
2 19. Sketch the graph of f ( x) 16 6 x x by completing the square. Ans:
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20. f(x) = x2 - 7 and g(x) = x4 + 5. Find f(g(x)). Ans: x8 + 10x4 + 18 21. Is f(x) = (x + 11)3 an even function, an odd function, or neither? Ans: neither 22. Determine whether the graph y = –3x2 - 6 is symmetric about the x-axis, the y-axis, or the origin. Ans: Symmetric about the y-axis 23. Determine whether the graph y = –3x3 - 8x is symmetric about the x-axis, the y-axis, or the origin. Ans: The graph is symmetric about the origin. 24. Determine whether the graph x5 = –7y5 - 5y is symmetric about the x-axis, the y-axis, or the origin. Ans: The graph is symmetric about the origin. 25. Find all intercepts of 4x2-y2 = 20 and determine symmetry about the x-axis, the y-axis, or the origin. Ans: x-intercepts: x 5 y-intercepts: none The graph is symmetric about both the x-axis and y-axis, it is symmetric about the origin.
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y
–9 8 x x3 and determine symmetry about the x-axis, the y-axis,
26. Find all intercepts of or the origin. Ans: x-intercepts: none y-intercepts: none The graph is symmetric about the origin.
27. Find all intercepts of x = 5y4 - 25y2 and determine symmetry about the x-axis, the y-axis, or the origin. Ans: x-intercept: x = 0 y-intercept: y 5 and y = 0 The graph is symmetric about the x-axis. 28. Find all intercepts of y4 = |2x| + 4 and determine symmetry about the x-axis, the y-axis, or the origin. Ans: x-intercept: none 4 y-intercept: y 4 The graph is symmetric about both the x and y-axes. The graph is symmetric about the origin. 29. Find all intercepts of y9 = |9x| - 7 and determine symmetry about the x-axis, the y-axis, or the origin. 7 x 9 Ans: x-intercept: 9 y-intercept: y 7 The graph is symmetric about the y-axis.
30. Given f(x) = x2 + 14x + 40 is not a one to one function. Modify the domain of f so that it will be a one-to-one function. , –7 or on the range –7, Ans: f ( x) is one-to-one on the range
2 31 Given that f ( x) 9 x is not a one-to-one function. Modify the domain of f so that it will be a one-to-one function. Ans: f(x) is one-to-one on [-3, 0] or on [0, 3]
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32. Find f -1(2) if
f ( x) –11x5 5
Ans: f -1(2) =
–
7 8 .
9 88 f ( x)
10 x 7 2 x 10 . What does this result tell you about the graph of f?
33. Find the inverse of 10 x 7 f 1 ( x) 2 x 10 Ans: The graph of f is symmetric about the line y = x. 4 34. Find f -1(x) if f ( x) 8x 5 . x4 5 f 1 ( x) 8 Ans:
35. Determine whether or not f(x) = 2x5+9x3 + 6x - 7 is a one-to-one function. Ans: Yes, f(x) is a one -to-one function.
g ( x)
36. Determine if f(x) = 9x + 2 and Ans: Yes
x2 9 are inverses of each other.
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