Inverse Functions and Relations

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