1330_Ch1_Section5

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:



Solution:



The Inverse of a One-to-One Function:

Example:

Solution:



A Technique for Finding the Inverse of a One-to-One

Function:

Example:



Solution:

The Graphs of f and f -1:

Example:



Solution:

Additional Example 1:



Solution:

Exercise Set 1.5: Inverse Functions


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



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


10 1f , find 10 .g


9 3f , find 3 .g f


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



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 ; 14

1g 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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1330_Ch1_Section5