Inverse Functions

1. Section 3-8
Inverse Functions

2. Warm-up
Indicate how you would “undo” each operation or
composite of operations.
1. Turn east and walk 50 meters, then turn north and
walk 30 meters.
4
2. Multiply a number by .
5
3. Add -70 to a number, then multiply the result by
14.
4. Square a positive number, then cube it.

3. Inverse of a function:

4. Inverse of a function:
A function that will “undo” what another function
had previously done

5. Inverse of a function:
A function that will “undo” what another function
had previously done
When the independent variable is switched with the
dependent variable

6. Inverse of a function:
A function that will “undo” what another function
had previously done
When the independent variable is switched with the
dependent variable
**Notation: The inverse of f is f-1

7. Example 1
Let S = {(1,1), (2, 4), (3, 9), (4, 16)}.
a. Find the inverse S-1.
b. Describe S and its inverse in words.

8. Example 1
Let S = {(1,1), (2, 4), (3, 9), (4, 16)}.
a. Find the inverse S-1.
S-1 = {(1,1), (4, 2), (9, 3), (16, 4)}
b. Describe S and its inverse in words.

9. Example 1
Let S = {(1,1), (2, 4), (3, 9), (4, 16)}.
a. Find the inverse S-1.
S-1 = {(1,1), (4, 2), (9, 3), (16, 4)}
b. Describe S and its inverse in words.
S is a squaring function, where the independent
variable is squared to obtain the dependent variable.

10. Example 1
Let S = {(1,1), (2, 4), (3, 9), (4, 16)}.
a. Find the inverse S-1.
S-1 = {(1,1), (4, 2), (9, 3), (16, 4)}
b. Describe S and its inverse in words.
S is a squaring function, where the independent
variable is squared to obtain the dependent variable.
Its inverse is a positive square root function, where
you would square root the independent variable to
get the dependent variable.

11. Just a little note:

12. Just a little note:
As the independent variable switches with the dependent variable, the domain switches with the range.

13. Theorem
(Horizontal-Line Test)

14. Theorem
(Horizontal-Line Test)
If you can draw a horizontal line on the graph of f and
it touches the graph more than once, then the
INVERSE of f is not a function.

15. Theorem
(Horizontal-Line Test)
If you can draw a horizontal line on the graph of f and
it touches the graph more than once, then the
INVERSE of f is not a function.
The horizontal-line test tells us nothing about the
original function...remember that!

16. Example 2
Give an equation for the inverse and tell whether it is
a function.
a. f ( x ) = 6x + 5

17. Example 2
Give an equation for the inverse and tell whether it is
a function.
a. f ( x ) = 6x + 5
y = 6x + 5

18. Example 2
Give an equation for the inverse and tell whether it is
a function.
a. f ( x ) = 6x + 5
y = 6x + 5
x = 6y + 5

19. Example 2
Give an equation for the inverse and tell whether it is
a function.
a. f ( x ) = 6x + 5
y = 6x + 5
x = 6y + 5
−5 −5

20. Example 2
Give an equation for the inverse and tell whether it is
a function.
a. f ( x ) = 6x + 5
y = 6x + 5
x = 6y + 5
−5 −5
x − 5 = 6y

21. Example 2
Give an equation for the inverse and tell whether it is
a function.
a. f ( x ) = 6x + 5
y = 6x + 5
x = 6y + 5
−5 −5
x − 5 = 6y
x−5
y=
6

22. Example 2
Give an equation for the inverse and tell whether it is
a function.
a. f ( x ) = 6x + 5
y = 6x + 5
x = 6y + 5
−5 −5
x − 5 = 6y
x−5
y= or
6

23. Example 2
Give an equation for the inverse and tell whether it is
a function.
a. f ( x ) = 6x + 5
y = 6x + 5
x = 6y + 5
−5 −5
x − 5 = 6y
x−5
y= or
6
1 5
y= x−
6 6

24. Example 2
4
b. y =
3x − 1

25. Example 2
4
b. y =
3x − 1
4
x=
3y − 1

26. Example 2
4
b. y =
3x − 1
4
x=
3y − 1
4
3y − 1 =
x

27. Example 2
4
b. y =
3x − 1
4
x=
3y − 1
4
3y − 1 =
x
4
3y = + 1
x

28. Example 2
4
b. y =
3x − 1
4
x=
3y − 1
4
3y − 1 =
x
4
3y = + 1
x

29. Example 2
4
b. y =
3x − 1
4
x=
3y − 1
4
3y − 1 =
x
4 4x
3y = + 1 3y = +
xx
x

30. Example 2
4
b. y =
3x − 1
4
x=
3y − 1
4
3y − 1 =
x
4 4x
3y = + 1 3y = +
xx
x

31. Example 2
4
b. y =
3x − 1
4
x=
3y − 1
4
3y − 1 =
x
4+x
4 4x
3y = + 1 3y =
3y = +
xx x
x

32. Example 2
4
b. y =
3x − 1
4
x=
3y − 1
4
3y − 1 =
x
4+x
4 4x
3y = + 1 3y =
3y = +
xx x
x
4+x
y=
3x

33. Question:
How do you verify that two functions are inverses of
each other?

34. Question:
How do you verify that two functions are inverses of
each other?
Use the Inverse Function Theorem!

35. Question:
How do you verify that two functions are inverses of
each other?
Use the Inverse Function Theorem!
The IFT says that two functions f and g are inverses
of each other IFF f(g(x)) = x for all x in the domain of
g AND g(f(x)) = x for all x in the domain of f.

36. Example 3
Verify that the functions in Example 2a are inverses of
each other.

37. Example 3
Verify that the functions in Example 2a are inverses of
each other.
To do this, we have to show that f(g(x)) = x and
g(f(x)) = x.

38. Example 3
Verify that the functions in Example 2a are inverses of
each other.
To do this, we have to show that f(g(x)) = x and
g(f(x)) = x.
Let’s calculate this together.

39. Example 4
Explain why the functions f and g, with f(m) = m2 and
g(m) = m-2 are not inverses.

40. Example 4
Explain why the functions f and g, with f(m) = m2 and
g(m) = m-2 are not inverses.
Calculate f(g(m)). If this composite does not give us a
value of m, then we know they are not inverses. If it
does, then we have to check g(f(m)).

41. Homework

42. Homework
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