1. Inverse Functions

2. DEFINITION Two relations are inverses if and only if when one relation contains (a,b), the other relation contains (b,a).

3. EXAMPLES Suppose and. Find: 1. 2. 3.

4. EXAMPLES Suppose and. Find: 1. 2. 3. 4 3 6

5. IMPORTANT IDEA Did you notice that : and In general: and

6. EXAMPLE Let The inverse of f(x) is: The graph of f(x) and f -1 (x) looks like…

7. IMPORTANT IDEA The last example leads us to: f -1 (x) is a reflection of f(x) over the line y = x.

8. FINDING AN INVERSE Given a function, find its inverse by: 1. Letting y = f(x) 2. Interchange x and y 3. Solve for the new y 4. Rename y as f -1 (x) {only if it is a function)

9. EXAMPLE Find the inverse of the following functions: 1. 2.

10. EXAMPLE Find the inverse of the following functions: 1. 2.

11. EXAMPLES CONTINUED 3.4.

12. EXAMPLES CONTINUED 3.4.

13. IMPORTANT IDEA The inverse of a function may or may not be a function. For example:

14. HORIZONTAL LINE TEST If any horizontal line intersects the graph of a function in no more than one point, its inverse is a function.

15. DEFINITION If the original and the inverse are both functions, then it is said to be one-to-one. No x-values AND no y-values repeat!

16. ASSIGNMENT RED BOOK Page 267-269 Problems 9-21odd, 31-39odd, and 47-53odd