The document discusses inverse functions and relations. It defines an inverse relation as one where the coordinates of a relation are switched, and an inverse function as one where the domain and range of a function are switched. It provides examples of finding the inverse of specific relations and functions by switching their coordinates or domain and range. It also discusses how to determine if two functions are inverses using their graphs and the horizontal line test.

1. Section 5-3
Inverse Functions and Relations

2. Essential Questions
• How do you find the inverse of a function or
relation?
• How do you determine whether two functions
or relations are inverses?

3. Vocabulary
1. Inverse Relation:
2. Inverse Function:

4. Vocabulary
1. Inverse Relation: When the coordinates of a
relation are switched
2. Inverse Function:

5. Vocabulary
1. Inverse Relation: When the coordinates of a
relation are switched
Two relations are inverse relations IFF one
relation contains (a, b) and the other relation
contains (b, a).
2. Inverse Function:

6. Vocabulary
1. Inverse Relation: When the coordinates of a
relation are switched
Two relations are inverse relations IFF one
relation contains (a, b) and the other relation
contains (b, a).
2. Inverse Function: When the domain and range
of one function are switched to form a new
function

7. Vocabulary
1. Inverse Relation: When the coordinates of a
relation are switched
Two relations are inverse relations IFF one
relation contains (a, b) and the other relation
contains (b, a).
2. Inverse Function: When the domain and range
of one function are switched to form a new
function
If f and f-1 are inverses, then f(a) = b IFF
f-1(b) = a

8. Vocabulary
3. Horizontal Line Test:

9. Vocabulary
3. Horizontal Line Test: Will test if the inverse of a
function is also a function

10. Vocabulary
3. Horizontal Line Test: Will test if the inverse of a
function is also a function
A function f has an inverse function f-1 IFF
each horizontal line intersects the graph of
the function in at most one point

11. Example 1
{(1,3),(6,3),(6,0),(1,0)}
The ordered pairs of the relation
are the coordinates of the vertices of a
rectangle. Find the inverse of this relation.
Describe the graph of the inverse.

12. Example 1
{(1,3),(6,3),(6,0),(1,0)}
The ordered pairs of the relation
are the coordinates of the vertices of a
rectangle. Find the inverse of this relation.
Describe the graph of the inverse.
{(3,1),(3,6),(0,6),(0,1)}

13. Example 1
{(1,3),(6,3),(6,0),(1,0)}
The ordered pairs of the relation
are the coordinates of the vertices of a
rectangle. Find the inverse of this relation.
Describe the graph of the inverse.
{(3,1),(3,6),(0,6),(0,1)}
The points in the inverse are still the vertices of a
rectangle, but reflected across the line y = x.

14. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
x
y

15. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x
y

16. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x
y

17. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
x
y

18. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
x
y

19. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

20. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

21. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

22. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

23. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

24. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

25. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

26. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

27. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

28. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

29. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

30. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

31. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

32. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

33. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

34. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

35. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

36. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

37. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

38. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

39. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

40. Example 2
Find the inverse of the function below. Graph
the original function and its inverse.
f (x ) = −
1
2
x +1
y = −
1
2
x +1
x = −
1
2
y +1
x −1= −
1
2
y
−2x + 2 = y
f −1
(x ) = −2x + 2 x
y

41. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1

42. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
y = x 2
− 4x +1

43. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
y = x 2
− 4x +1
x = y 2
− 4y +1

44. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
y = x 2
− 4x +1
x = y 2
− 4y +1
x −1= y 2
− 4y

45. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
y = x 2
− 4x +1
x = y 2
− 4y +1
x −1= y 2
− 4y
1
2
i −4
⎛
⎝⎜
⎞
⎠⎟
2

46. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
y = x 2
− 4x +1
x = y 2
− 4y +1
x −1= y 2
− 4y
1
2
i −4
⎛
⎝⎜
⎞
⎠⎟
2
−2( )2

47. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
y = x 2
− 4x +1
x = y 2
− 4y +1
x −1= y 2
− 4y
1
2
i −4
⎛
⎝⎜
⎞
⎠⎟
2
−2( )2
4

48. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
y = x 2
− 4x +1
x = y 2
− 4y +1
x −1= y 2
− 4y
1
2
i −4
⎛
⎝⎜
⎞
⎠⎟
2
−2( )2
4
x −1+ 4 = y 2
− 4y + 4

49. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
y = x 2
− 4x +1
x = y 2
− 4y +1
x −1= y 2
− 4y
1
2
i −4
⎛
⎝⎜
⎞
⎠⎟
2
−2( )2
4
x −1+ 4 = y 2
− 4y + 4
x + 3 = y 2
− 4y + 4

50. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
y = x 2
− 4x +1
x = y 2
− 4y +1
x −1= y 2
− 4y
1
2
i −4
⎛
⎝⎜
⎞
⎠⎟
2
−2( )2
4
x −1+ 4 = y 2
− 4y + 4
x + 3 = y 2
− 4y + 4
x + 3 = (y − 2)2

51. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
y = x 2
− 4x +1
x = y 2
− 4y +1
x −1= y 2
− 4y
1
2
i −4
⎛
⎝⎜
⎞
⎠⎟
2
−2( )2
4
x −1+ 4 = y 2
− 4y + 4
x + 3 = y 2
− 4y + 4
x + 3 = (y − 2)2
± x + 3 = (y − 2)2

52. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
y = x 2
− 4x +1
x = y 2
− 4y +1
x −1= y 2
− 4y
1
2
i −4
⎛
⎝⎜
⎞
⎠⎟
2
−2( )2
4
x −1+ 4 = y 2
− 4y + 4
x + 3 = y 2
− 4y + 4
x + 3 = (y − 2)2
± x + 3 = (y − 2)2
± x + 3 = y − 2

53. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
y = x 2
− 4x +1
x = y 2
− 4y +1
x −1= y 2
− 4y
1
2
i −4
⎛
⎝⎜
⎞
⎠⎟
2
−2( )2
4
x −1+ 4 = y 2
− 4y + 4
x + 3 = y 2
− 4y + 4
x + 3 = (y − 2)2
± x + 3 = (y − 2)2
± x + 3 = y − 2
2 ± x + 3 = y

54. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
y = x 2
− 4x +1
x = y 2
− 4y +1
x −1= y 2
− 4y
1
2
i −4
⎛
⎝⎜
⎞
⎠⎟
2
−2( )2
4
x −1+ 4 = y 2
− 4y + 4
x + 3 = y 2
− 4y + 4
x + 3 = (y − 2)2
± x + 3 = (y − 2)2
± x + 3 = y − 2
2 ± x + 3 = y
f −1
= 2 ± x + 3

55. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x
y

56. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x
y

57. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x
y

58. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
x
y

59. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
f −1
= 2 + x + 3; (−∞,2]
x
y

60. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
f −1
= 2 + x + 3; (−∞,2]
f −1
= 2 − x + 3; [2,+∞)
x
y

61. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
f −1
= 2 + x + 3; (−∞,2]
f −1
= 2 − x + 3; [2,+∞)
x
y

62. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
f −1
= 2 + x + 3; (−∞,2]
f −1
= 2 − x + 3; [2,+∞)
x
y

63. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
f −1
= 2 + x + 3; (−∞,2]
f −1
= 2 − x + 3; [2,+∞)
x
y

64. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
f −1
= 2 + x + 3; (−∞,2]
f −1
= 2 − x + 3; [2,+∞)
x
y

65. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
f −1
= 2 + x + 3; (−∞,2]
f −1
= 2 − x + 3; [2,+∞)
x
y

66. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
f −1
= 2 + x + 3; (−∞,2]
f −1
= 2 − x + 3; [2,+∞)
x
y

67. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
f −1
= 2 + x + 3; (−∞,2]
f −1
= 2 − x + 3; [2,+∞)
x
y

68. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
f −1
= 2 + x + 3; (−∞,2]
f −1
= 2 − x + 3; [2,+∞)
x
y

69. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
f −1
= 2 + x + 3; (−∞,2]
f −1
= 2 − x + 3; [2,+∞)
x
y

70. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
f −1
= 2 + x + 3; (−∞,2]
f −1
= 2 − x + 3; [2,+∞)
x
y

71. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
f −1
= 2 + x + 3; (−∞,2]
f −1
= 2 − x + 3; [2,+∞)
x
y

72. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
f −1
= 2 + x + 3; (−∞,2]
f −1
= 2 − x + 3; [2,+∞)
x
y

73. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
f −1
= 2 + x + 3; (−∞,2]
f −1
= 2 − x + 3; [2,+∞)
x
y

74. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
f −1
= 2 + x + 3; (−∞,2]
f −1
= 2 − x + 3; [2,+∞)
x
y

75. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
f −1
= 2 + x + 3; (−∞,2]
f −1
= 2 − x + 3; [2,+∞)
x
y

76. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
f −1
= 2 + x + 3; (−∞,2]
f −1
= 2 − x + 3; [2,+∞)
x
y

77. Example 3
Find the inverse of the function below. Then
graph the function and its inverse. If necessary,
restrict the domain of the inverse so that it is a
function.
f (x ) = x 2
− 4x +1
f −1
= 2 ± x + 3
x =
−b
2a
x =
4
2
x = 2
f −1
= 2 + x + 3; (−∞,2]
f −1
= 2 − x + 3; [2,+∞)
x
y

78. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.

79. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
Two functions f and g are inverse functions IFF both
of their compositions are the identity function.

80. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
Two functions f and g are inverse functions IFF both
of their compositions are the identity function.
f(x) and g(x) are inverses IFF
[f !g](x ) = x and [g !f ](x ) = x

81. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
Two functions f and g are inverse functions IFF both
of their compositions are the identity function.
f(x) and g(x) are inverses IFF
[f !g](x ) = x and [g !f ](x ) = x
f (x ) =
3
4
x − 6

82. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
Two functions f and g are inverse functions IFF both
of their compositions are the identity function.
f(x) and g(x) are inverses IFF
[f !g](x ) = x and [g !f ](x ) = x
f (x ) =
3
4
x − 6 g(x ) =
4
3
x + 8

83. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
f (x ) =
3
4
x − 6 g(x ) =
4
3
x + 8

84. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
f (x ) =
3
4
x − 6 g(x ) =
4
3
x + 8
[f ! g](x )

85. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
f (x ) =
3
4
x − 6 g(x ) =
4
3
x + 8
[f ! g](x )
= f [g(x )]

86. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
f (x ) =
3
4
x − 6 g(x ) =
4
3
x + 8
[f ! g](x )
= f [g(x )]
= f
4
3
x + 8
⎛
⎝⎜
⎞
⎠⎟

87. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
f (x ) =
3
4
x − 6 g(x ) =
4
3
x + 8
[f ! g](x )
= f [g(x )]
= f
4
3
x + 8
⎛
⎝⎜
⎞
⎠⎟
=
3
4
4
3
x + 8
⎛
⎝⎜
⎞
⎠⎟ − 6

88. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
f (x ) =
3
4
x − 6 g(x ) =
4
3
x + 8
[f ! g](x )
= f [g(x )]
= f
4
3
x + 8
⎛
⎝⎜
⎞
⎠⎟
=
3
4
4
3
x + 8
⎛
⎝⎜
⎞
⎠⎟ − 6
= x + 6 − 6

89. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
f (x ) =
3
4
x − 6 g(x ) =
4
3
x + 8
[f ! g](x )
= f [g(x )]
= f
4
3
x + 8
⎛
⎝⎜
⎞
⎠⎟
=
3
4
4
3
x + 8
⎛
⎝⎜
⎞
⎠⎟ − 6
= x + 6 − 6
= x

90. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
f (x ) =
3
4
x − 6 g(x ) =
4
3
x + 8
[f ! g](x )
= f [g(x )]
= f
4
3
x + 8
⎛
⎝⎜
⎞
⎠⎟
=
3
4
4
3
x + 8
⎛
⎝⎜
⎞
⎠⎟ − 6
= x + 6 − 6
= x
[f ! g](x ) = x

91. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
f (x ) =
3
4
x − 6 g(x ) =
4
3
x + 8

92. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
f (x ) =
3
4
x − 6 g(x ) =
4
3
x + 8
[g !f ](x )

93. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
f (x ) =
3
4
x − 6 g(x ) =
4
3
x + 8
[g !f ](x )
= g[f (x )]

94. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
f (x ) =
3
4
x − 6 g(x ) =
4
3
x + 8
[g !f ](x )
= g[f (x )]
= g
3
4
x − 6
⎛
⎝⎜
⎞
⎠⎟

95. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
f (x ) =
3
4
x − 6 g(x ) =
4
3
x + 8
[g !f ](x )
= g[f (x )]
= g
3
4
x − 6
⎛
⎝⎜
⎞
⎠⎟
=
4
3
3
4
x − 6
⎛
⎝⎜
⎞
⎠⎟ + 8

96. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
f (x ) =
3
4
x − 6 g(x ) =
4
3
x + 8
[g !f ](x )
= g[f (x )]
= g
3
4
x − 6
⎛
⎝⎜
⎞
⎠⎟
=
4
3
3
4
x − 6
⎛
⎝⎜
⎞
⎠⎟ + 8
= x − 8 + 8

97. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
f (x ) =
3
4
x − 6 g(x ) =
4
3
x + 8
[g !f ](x )
= g[f (x )]
= g
3
4
x − 6
⎛
⎝⎜
⎞
⎠⎟
=
4
3
3
4
x − 6
⎛
⎝⎜
⎞
⎠⎟ + 8
= x − 8 + 8
= x

98. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
f (x ) =
3
4
x − 6 g(x ) =
4
3
x + 8
[g !f ](x )
= g[f (x )]
= g
3
4
x − 6
⎛
⎝⎜
⎞
⎠⎟
=
4
3
3
4
x − 6
⎛
⎝⎜
⎞
⎠⎟ + 8
= x − 8 + 8
= x
[g !f ](x ) = x

99. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
f (x ) =
3
4
x − 6 g(x ) =
4
3
x + 8

100. Example 4
Determine whether the functions below are inverse
functions. Explain your reasoning.
f (x ) =
3
4
x − 6 g(x ) =
4
3
x + 8
Since and , these functions
are inverses.
[f !g](x ) = x [g !f ](x ) = x