1. Section 4.6 – Inverse Functions

2. DOES an inverse function exist?
IF YES, you can find the inverse function.

3. The Existence of the Inverse of f(x)
IF for every x there is at most one y (function)
AND
IF for every y there is at most one x (one-to-one)
then an inverse function of f(x) exists.
The inverse function is denoted by

4. Graphical Existence of Inverse
Passes BOTH vertical and horizontal line test.
Inverse Exists
No Inverse Exists
(1, 1), (-1, 1)
No Inverse Exists
(1, 0), (0, 0)

5. Inverse Exists
No Inverse Exists
(3, 0.9), (7, 0.9)

6. Inverse Exists
Inverse Exists

7. No Inverse Exists
(-3, 2), (0, 2)
No Inverse Exists
(2, -2), (2, 5)

8. (Jan, Winter), (Feb, Winter)
Does an inverse exist?
January
February
March
July
Winter
Spring
Summer
No Inverse Exists
(1, 2), (2, 2)
No Inverse Exists
(Jan, Winter), (Feb, Winter)
Ford
Bush
Carter
Clinton
President
Vice-President
No Inverse Exists
(2, 4), (3, 4)
No Inverse Exists
(Ford, President),
(Ford, Vice-President)
No Inverse Exists
(6, 5), (6, 3)

9. Algebraic Existence of Inverse
No Inverse Exists
(-4, -12), (-2, -12)
No Inverse Exists
(4, 2), (4, -2)
Inverse Exists
No Inverse Exists
(0, 2), (0, -2)
No Inverse Exists
(4, 0), (-4, 0)

10. FINDING the inverse which exists
1. Switch the x and the y.
2. (Algebraically) Solve for y.
3. Replace y with

11. Finding the inverse function GRAPHICALLY

13. Finding the Inverse Function TABULARLY
No Inverse Exists
(5, 4), (7, 4)

14. Finding the Inverse ANALYTICALLY