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CHAPTER 1 A Review of Functions
University of Houston Department of Mathematics 122
Section 1.5: Inverse Functions
Inverses of One-to-One Functions
Inverses of One-to-One Functions
Definition of a One-to-One Function:
Example:
Solution:
SECTION 1.5 Inverse Functions
MATH 1330 Precalculus 123
Example:
Solution:
Horizontal Line Test:
Example:
CHAPTER 1 A Review of Functions
University of Houston Department of Mathematics 124
Solution:
SECTION 1.5 Inverse Functions
MATH 1330 Precalculus 125
The Inverse of a One-to-One Function:
Example:
Solution:
CHAPTER 1 A Review of Functions
University of Houston Department of Mathematics 126
A Technique for Finding the Inverse of a One-to-One
Function:
Example:
SECTION 1.5 Inverse Functions
MATH 1330 Precalculus 127
Solution:
The Graphs of f and f -1:
Example:
CHAPTER 1 A Review of Functions
University of Houston Department of Mathematics 128
Solution:
Additional Example 1:
SECTION 1.5 Inverse Functions
MATH 1330 Precalculus 129
Solution:
CHAPTER 1 A Review of Functions
University of Houston Department of Mathematics 130
Additional Example 2:
Solution:
SECTION 1.5 Inverse Functions
MATH 1330 Precalculus 131
CHAPTER 1 A Review of Functions
University of Houston Department of Mathematics 132
SECTION 1.5 Inverse Functions
MATH 1330 Precalculus 133
Additional Example 3:
Solution:
CHAPTER 1 A Review of Functions
University of Houston Department of Mathematics 134
SECTION 1.5 Inverse Functions
MATH 1330 Precalculus 135
Additional Example 4:
Solution:
Additional Example 5:
CHAPTER 1 A Review of Functions
University of Houston Department of Mathematics 136
Solution:
Additional Example 6:
SECTION 1.5 Inverse Functions
MATH 1330 Precalculus 137
Solution:
Exercise Set 1.5: Inverse Functions
University of Houston Department of Mathematics 138
x
y
x
y
x
y
x
y
x
y
x
y
Determine whether each of the following graphs
represents a one-to-one function. Explain your answer.
1.
2.
3.
4.
5.
6.
For each of the following functions, sketch a graph
and then determine whether the function is one-to-one.
7. 2 3f x x
8. 2 5g x x
9. 3
2h x x
10. 3 2f x x
11. 4g x x
12. 1
3h xx
13. 2
2 1f x x
14. 6g x x
Answer the following.
15. If a function f is one-to-one, then the inverse
function, 1f , can be graphed by either of the
following methods:
(a) Interchange the ____ and ____ values.
(b) Reflect the graph of f over the line .____y
16. The domain of f is equal to the __________ of
1f , and the range of f is equal to the
__________ of .1f
A table of values for a one-to-one function y f x is
given. Complete the table for 1y f x
.
17.
18.
x f x
3 4
2 7
4 5
5 0
0 3
x 1f x
4
2
5
0
x f x
5 9
4 5
6 3
8 2
2 6
x 1f x
5
5
6
8
Exercise Set 1.5: Inverse Functions
MATH 1330 Precalculus 139
x
y
f
x
y
f
x
y
f
x
y
f
For each of the following graphs:
(a) State the domain and range of f .
(b) Sketch 1f .
(c) State the domain and range of 1
f
.
19.
20.
21.
22.
Answer the following. Assume that f is a one-to-one
function.
23. 1If 4 5, find 5 .f f
24. 1If 6 2, find 2 .f f
25. 1If 3 7, find 7 .f f
26. 1If 6 8, find 8 .f f
27. 1If 3 9 and (9) 5, find 9 .f f f
28. 1If 5 4 and (2) 5, find 5 .f f f
29. 1If 4 2, find 2 .f f f
30. 1 1If 5 3, find 3 .f f f
Answer the following. Assume that f and g are defined
for all real numbers.
31. If f and g are inverse functions, 2 3f and
4 2f , find 2 .g
32. If f and g are inverse functions, 7 10f and
10 1f , find 10 .g
33. If f and g are inverse functions, 5 8f and
9 3f , find 3 .g f
34. If f and g are inverse functions, 1 6f and
7 8f , find 6 .f g
For each of the following functions, write an equation
for the inverse function 1y f x
.
35. 5 3f x x
36. 4 7f x x
37. 3 2
8
xf x
38. 6 5
4
xf x
Exercise Set 1.5: Inverse Functions
University of Houston Department of Mathematics 140
39. 2 1f x x , where 0x
40. 25f x x , where 0x
41. 34 7f x x
42. 32 1f x x
43. 3
2f x
x
44. 5
7f x
x
45. 2 3
4
xf x
x
46. 3 8
5
xf x
x
47. 7 2f x x
48. 2 6 5f x x
Use the Property of Inverse Functions to determine
whether each of the following pairs of functions are
inverses of each other. Explain your answer.
49. 4 1f x x ; 1
41g x x
50. 2 3f x x ; 2
3
xg x
51. 4
5
xf x
; 4 5g x x
52. 2 5f x x ; 1
2 5g x
x
53. 3 2f x x ; 3 2g x x
54. 5 7f x x ; 5
7g x x
55. 5
f xx
; 5
g xx
56. 2 9f x x , where 0x ;
9g x x
Answer the following.
57. If f x is a function that represents the amount
of revenue (in dollars) by selling x tickets, then
what does 1 500f represent?
58. If f x is a function that represents the area of a
circle with radius x, then what does 1 80f
represent?
A function is said to be one-to-one provided that the
following holds for all 1x and 2x in the domain of f :
If 1 2f x f x , then 1 2x x .
Use the above definition to determine whether or not
the following functions are one-to-one. If f is not one-
to-one, then give a specific example showing that the
condition 1 2f x f x fails to imply that 1 2x x .
59. 5 3f x x
60. 3 5f x x
61. 4f x x
62. 4f x x
63. 4f x x
64. 1
4f xx
65. 2 3f x x
66. 2
3f x x
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