1. Section 9.1 – 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

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

8. January February March July Winter Spring Summer Does an inverse exist? Ford Bush Carter Clinton President Vice-President No Inverse Exists (Jan, Winter), (Feb, Winter) No Inverse Exists (1, 2), (2, 2) Inverse Exists 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 2. Switch the x and the y. 3. (Algebraically) Solve for y. 4. Replace y with 1. Determine if inverse function exists. If yes, proceed to #2.

11. Finding the inverse function GRAPHICALLY

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

14. Finding the Inverse ANALYTICALLY