Properties of Inverse Functions

1. Section 8-3
Properties of Inverse Functions

2. Inverse Functions

3. Inverse Functions
1. Find them by switching x and y

4. Inverse Functions
1. Find them by switching x and y
2. A graph and its inverse are reflections over the line
y=x

5. Inverse Functions
1. Find them by switching x and y
2. A graph and its inverse are reflections over the line
y=x
3. The domain of g = the range of f; the range of g =
the domain of f

6. What is an inverse?

7. What is an inverse?
An inverse is something that “undoes” something that was
already done.

8. What is an inverse?
An inverse is something that “undoes” something that was
already done.
Start with 3. Add 4 to get 7. To go back to 3, subtract 4,
which is the inverse of adding 4.

9. What are inverse functions?

10. Inverse Function Theorem
Two functions f and g are inverse functions IFF:

11. Inverse Function Theorem
Two functions f and g are inverse functions IFF:
1. For all x in the domain of f, g o f(x) = x

12. Inverse Function Theorem
Two functions f and g are inverse functions IFF:
1. For all x in the domain of f, g o f(x) = x
2. For all x in the domain of g, f o g(x) = x

13. Inverse Function Theorem
Two functions f and g are inverse functions IFF:
1. For all x in the domain of f, g o f(x) = x
2. For all x in the domain of g, f o g(x) = x
When applying inverse functions, when you start with a
number “x” and apply one and then the other, you will
end up back at x, regardless of the order you take the
functions.

14. Example 1
1
Let f : x → x − 3
2
a. Find the inverse. Call it g.

15. Example 1
1
Let f : x → x − 3
2
a. Find the inverse. Call it g.
1
y = x−3
2

16. Example 1
1
Let f : x → x − 3
2
a. Find the inverse. Call it g.
1
y = x−3
2
1
x= y−3
2

17. Example 1
1
Let f : x → x − 3
2
a. Find the inverse. Call it g.
1
y = x−3
2
1
x= y−3
2
+3 +3

18. Example 1
1
Let f : x → x − 3
2
a. Find the inverse. Call it g.
1
y = x−3
2
1
x= y−3
2
+3 +3
1
x+3= y 2

19. Example 1
1
Let f : x → x − 3
2
a. Find the inverse. Call it g.
1
y = x−3
2
1
x= y−3
2
+3 +3
1
x+3= y 2
1
2(x + 3) = ( y)22

20. Example 1
1
Let f : x → x − 3
2
a. Find the inverse. Call it g.
1
y = x−3
2
1
x= y−3
2
+3 +3
1
x+3= y 2
1
2(x + 3) = ( y)22
y = 2x + 6

21. Example 1
1
Let f : x → x − 3
2
a. Find the inverse. Call it g.
1
y = x−3
2
1
x= y−3
2
+3 +3
1
x+3= y 2
1
2(x + 3) = ( y)22
y = 2x + 6
g(x) = 2x + 6

22. Example 1
b. Find f(g(x))
1
f(x) = x − 3 g(x) = 2x + 6
2

23. Example 1
b. Find f(g(x))
1
f(x) = x − 3 g(x) = 2x + 6
2
f( g(x))

24. Example 1
b. Find f(g(x))
1
f(x) = x − 3 g(x) = 2x + 6
2
f( g(x))
= f(2x + 6)

25. Example 1
b. Find f(g(x))
1
f(x) = x − 3 g(x) = 2x + 6
2
f( g(x))
= f(2x + 6)
1
= (2x + 6) − 3
2

26. Example 1
b. Find f(g(x))
1
f(x) = x − 3 g(x) = 2x + 6
2
f( g(x))
= f(2x + 6)
1
= (2x + 6) − 3
2
= x+3−3

27. Example 1
b. Find f(g(x))
1
f(x) = x − 3 g(x) = 2x + 6
2
f( g(x))
= f(2x + 6)
1
= (2x + 6) − 3
2
= x+3−3
=x

28. Example 1
c. Find g(f(x))
1
f(x) = x − 3 g(x) = 2x + 6
2

29. Example 1
c. Find g(f(x))
1
f(x) = x − 3 g(x) = 2x + 6
2
g( f(x))

30. Example 1
c. Find g(f(x))
1
f(x) = x − 3 g(x) = 2x + 6
2
g( f(x))
1
= g( x − 3)
2

31. Example 1
c. Find g(f(x))
1
f(x) = x − 3 g(x) = 2x + 6
2
g( f(x))
1
= g( x − 3)
2
1
= 2( x − 3) + 6
2

32. Example 1
c. Find g(f(x))
1
f(x) = x − 3 g(x) = 2x + 6
2
g( f(x))
1
= g( x − 3)
2
1
= 2( x − 3) + 6
2
= x−6+6

33. Example 1
c. Find g(f(x))
1
f(x) = x − 3 g(x) = 2x + 6
2
g( f(x))
1
= g( x − 3)
2
1
= 2( x − 3) + 6
2
= x−6+6
=x

34. Inverse Function Notation
For a function f(x), the inverse is f-1(x)

35. Example 2
1
f(t) = t − 3 f (t) = 4t + 12
−1
4
Find the inverse and check to make sure it is the inverse.

36. Example 2
1
f(t) = t − 3 f (t) = 4t + 12
−1
4
Find the inverse and check to make sure it is the inverse.
1
s= t−3
4

37. Example 2
1
f(t) = t − 3 f (t) = 4t + 12
−1
4
Find the inverse and check to make sure it is the inverse.
1
s= t−3
4
1
t = s−3
4

38. Example 2
1
f(t) = t − 3 f (t) = 4t + 12
−1
4
Find the inverse and check to make sure it is the inverse.
1
s= t−3
4
1
t = s−3
4
+3 +3

39. Example 2
1
f(t) = t − 3 f (t) = 4t + 12
−1
4
Find the inverse and check to make sure it is the inverse.
1
s= t−3
4
1
t = s−3
4
+3 +3
1
t+3= s 4

40. Example 2
1
f(t) = t − 3 f (t) = 4t + 12
−1
4
Find the inverse and check to make sure it is the inverse.
1
s= t−3
4
1
t = s−3
4
+3 +3
1
t+3= s 4
1
4(t + 3) = ( 4 s)4

41. Example 2
1
f(t) = t − 3 f (t) = 4t + 12
−1
4
Find the inverse and check to make sure it is the inverse.
1
s= t−3
4
1
t = s−3
4
+3 +3
1
t+3= s 4
1
4(t + 3) = ( 4 s)4
s = 4t + 12

42. Example 2
1
f(t) = t − 3 f (t) = 4t + 12
−1
4
Find the inverse and check to make sure it is the inverse.
1
s= t−3
4
1
t = s−3
4
+3 +3
1
t+3= s 4
1
4(t + 3) = ( 4 s)4
s = 4t + 12
f (t) = 4t + 12
−1

43. Example 2
1
f(t) = t − 3 f (t) = 4t + 12
−1
4
Find the inverse and check to make sure it is the inverse.
1
s= t−3 f( f (t)) = f(4t + 12)
−1
4
1
t = s−3
4
+3 +3
1
t+3= s 4
1
4(t + 3) = ( 4 s)4
s = 4t + 12
f (t) = 4t + 12
−1

44. Example 2
1
f(t) = t − 3 f (t) = 4t + 12
−1
4
Find the inverse and check to make sure it is the inverse.
1
s= t−3 f( f (t)) = f(4t + 12)
−1
4
1
= (4t + 12) − 3
1
t = s−3 4
4
+3 +3
1
t+3= s 4
1
4(t + 3) = ( 4 s)4
s = 4t + 12
f (t) = 4t + 12
−1

45. Example 2
1
f(t) = t − 3 f (t) = 4t + 12
−1
4
Find the inverse and check to make sure it is the inverse.
1
s= t−3 f( f (t)) = f(4t + 12)
−1
4
1
= (4t + 12) − 3
1
t = s−3 4
4
+3 +3
= t +3−3
1
t+3= s 4
1
4(t + 3) = ( 4 s)4
s = 4t + 12
f (t) = 4t + 12
−1

46. Example 2
1
f(t) = t − 3 f (t) = 4t + 12
−1
4
Find the inverse and check to make sure it is the inverse.
1
s= t−3 f( f (t)) = f(4t + 12)
−1
4
1
= (4t + 12) − 3
1
t = s−3 4
4
+3 +3
= t +3−3 = t
1
t+3= s 4
1
4(t + 3) = ( 4 s)4
s = 4t + 12
f (t) = 4t + 12
−1

47. Example 2
1
f(t) = t − 3 f (t) = 4t + 12
−1
4
Find the inverse and check to make sure it is the inverse.
1
s= t−3 f( f (t)) = f(4t + 12)
−1
4
1
= (4t + 12) − 3
1
t = s−3 4
4
+3 +3
= t +3−3 = t
1
t+3= s 4
−1 1
f ( f(t)) = f ( t − 3)
−1
1
4(t + 3) = ( 4 s)4 4
s = 4t + 12
f (t) = 4t + 12
−1

48. Example 2
1
f(t) = t − 3 f (t) = 4t + 12
−1
4
Find the inverse and check to make sure it is the inverse.
1
s= t−3 f( f (t)) = f(4t + 12)
−1
4
1
= (4t + 12) − 3
1
t = s−3 4
4
+3 +3
= t +3−3 = t
1
t+3= s 4
−1 1
f ( f(t)) = f ( t − 3)
−1
1
4(t + 3) = ( 4 s)4 4
1
= 4( t − 3) + 12
s = 4t + 12 4
f (t) = 4t + 12
−1

49. Example 2
1
f(t) = t − 3 f (t) = 4t + 12
−1
4
Find the inverse and check to make sure it is the inverse.
1
s= t−3 f( f (t)) = f(4t + 12)
−1
4
1
= (4t + 12) − 3
1
t = s−3 4
4
+3 +3
= t +3−3 = t
1
t+3= s 4
−1 1
f ( f(t)) = f ( t − 3)
−1
1
4(t + 3) = ( 4 s)4 4
1
= 4( t − 3) + 12
s = 4t + 12 4
= t − 12 + 12
f (t) = 4t + 12
−1

50. Example 2
1
f(t) = t − 3 f (t) = 4t + 12
−1
4
Find the inverse and check to make sure it is the inverse.
1
s= t−3 f( f (t)) = f(4t + 12)
−1
4
1
= (4t + 12) − 3
1
t = s−3 4
4
+3 +3
= t +3−3 = t
1
t+3= s 4
−1 1
f ( f(t)) = f ( t − 3)
−1
1
4(t + 3) = ( 4 s)4 4
1
= 4( t − 3) + 12
s = 4t + 12 4
= t − 12 + 12 = t
f (t) = 4t + 12
−1

51. Example 3
4
h(x) = x . Find h (x) and check.
−1

52. Example 3
4
h(x) = x . Find h (x) and check.
−1
4
y=x

53. Example 3
4
h(x) = x . Find h (x) and check.
−1
4
y=x
4
x= y

54. Example 3
4
h(x) = x . Find h (x) and check.
−1
4
y=x
4
x= y
1 1
4
x = (y )
4 4

55. Example 3
4
h(x) = x . Find h (x) and check.
−1
4
y=x
4
x= y
1 1
4
x = (y )
4 4
1
y=x 4

56. Example 3
4
h(x) = x . Find h (x) and check.
−1
4
y=x
4
x= y
1 1
4
x = (y )
4 4
1
y=x 4
1
h (x) = x when x ≥ 0
−1 4

57. Example 3
4
h(x) = x . Find h (x) and check.
−1
1
4
y=x h(h (x)) = h(x )
−1 4
4
x= y
1 1
4
x = (y )
4 4
1
y=x 4
1
h (x) = x when x ≥ 0
−1 4

58. Example 3
4
h(x) = x . Find h (x) and check.
−1
1
4
y=x h(h (x)) = h(x )
−1 4
4 1
x= y 4
= (x )4
1 1
4
x = (y )
4 4
1
y=x 4
1
h (x) = x when x ≥ 0
−1 4

59. Example 3
4
h(x) = x . Find h (x) and check.
−1
1
4
y=x h(h (x)) = h(x )
−1 4
4 1
=x
x= y 4
= (x )4
1 1
4
x = (y )
4 4
1
y=x 4
1
h (x) = x when x ≥ 0
−1 4

60. Example 3
4
h(x) = x . Find h (x) and check.
−1
1
4
y=x h(h (x)) = h(x )
−1 4
4 1
=x
x= y 4
= (x )
4
1 1
4
x = (y )
4 4
4
h (h(x)) = h (x )
−1 −1
1
y=x 4
1
h (x) = x when x ≥ 0
−1 4

61. Example 3
4
h(x) = x . Find h (x) and check.
−1
1
4
y=x h(h (x)) = h(x )
−1 4
4 1
=x
x= y 4
= (x )
4
1 1
4
x = (y )
4 4
4
h (h(x)) = h (x )
−1 −1
1 1
4
= (x )
y=x 4 4
1
h (x) = x when x ≥ 0
−1 4

62. Example 3
4
h(x) = x . Find h (x) and check.
−1
1
4
y=x h(h (x)) = h(x )
−1 4
4 1
=x
x= y 4
= (x )
4
1 1
4
x = (y )
4 4
4
h (h(x)) = h (x )
−1 −1
1 1
=x
4
= (x )
y=x 4 4
1
h (x) = x when x ≥ 0
−1 4

63. Power Function Inverse
Theorem
1
n
If f(x) = x and g(x) = x and the domains of f and g are
n
the set of nonnegative real numbers, then f and g are
inverse functions.

64. Example 4
Are the following inverses of each other?
1
g(x) = x − 2
f(x) = 3x + 2 3

65. Example 4
Are the following inverses of each other?
1
g(x) = x − 2
f(x) = 3x + 2 3
1
f( g(x)) = f( 3 x − 2)

66. Example 4
Are the following inverses of each other?
1
g(x) = x − 2
f(x) = 3x + 2 3
1
f( g(x)) = f( 3 x − 2)
1
= 3( x − 2) + 2
3

67. Example 4
Are the following inverses of each other?
1
g(x) = x − 2
f(x) = 3x + 2 3
1
f( g(x)) = f( 3 x − 2)
1
= 3( x − 2) + 2
3
= x−6+2

68. Example 4
Are the following inverses of each other?
1
g(x) = x − 2
f(x) = 3x + 2 3
1
f( g(x)) = f( 3 x − 2)
1
= 3( x − 2) + 2
3
= x−6+2
= x− 4

69. Example 4
Are the following inverses of each other?
1
g(x) = x − 2
f(x) = 3x + 2 3
1
f( g(x)) = f( 3 x − 2)
1
= 3( x − 2) + 2
3
= x−6+2
= x− 4
≠x

70. Example 4
Are the following inverses of each other?
1
g(x) = x − 2
f(x) = 3x + 2 3
1
f( g(x)) = f( 3 x − 2)
1
= 3( x − 2) + 2
3
= x−6+2
= x− 4
≠x
f(x) and g(x) are not inverses of each other

71. Homework

72. Homework
p. 493 #1-19
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