1. SECTION 4.2 ONE-TO-ONE FUNCTIONS ONE-TO-ONE FUNCTIONS INVERSE FUNCTIONS INVERSE FUNCTIONS

2. INVERSE FUNCTIONS There are some functions which we almost intuitively know as inverses of each other: There are some functions which we almost intuitively know as inverses of each other: Cubing a number, taking the cube root of a number. Cubing a number, taking the cube root of a number. Adding a value to a number, subtracting that value from the number. Adding a value to a number, subtracting that value from the number.

3. INVERSE FUNCTIONS These are inverses of each other because one undoes the other. Can a more complicated function have an inverse? Adds two and divides by 6.

4. What must the inverse of f(x) do to its variable, x? Multiply by 6 and subtract two.

5. x f(x) x g(x) x f(x) x g(x) 0 1 4 - 2 - 2 - 8 - 81/31/2 1 0 - 1 0 1 - 1 - 1 1/3 1/3 1/2 1/2 - 2 - 2 4 - 8 - 8 0 1 Symbolically? Numerically? Graphically?

6. ANOTHER IMPORTANT OBSERVATION  4 4 4 4  1  1  1  4 4 4 4 g(f(4)) = 4 In fact, the same thing happens for any x-value. g(f(x)) = x f(x) g(x)

7. EXAMPLE: Find the inverse of f(x) which we refer to as f -1 (x). Find the inverse of f(x) which we refer to as f -1 (x). Then, check algebraically to ensure that f(f -1 (x)) = x. Then, check algebraically to ensure that f(f -1 (x)) = x. f -1 (x) = (x - 6) 3 f -1 (x) = (x - 6) 3

8. = x - 6 + 6 = x - 6 + 6 = x = x Check that f (f -1 (x)) = x

9. RECALL: DEFINITION OF FUNCTION A set of ordered pairs in which no two ordered pairs have the same first coordinate. In other words: FOR EVERY X, THERE IS ONLY ONE Y.

10. Consider the function f(x) = x 2 Consider the function f(x) = x 2 x f(x) x f(x) 0 0 0 0 1 1 1 1 2 4 -1 1 -2 4 If this function had an inverse, the ordered pairs would have to be reversed. x y x y 0 0 0 0 1 1 1 1 4 2 4 2 1 -1 1 -1 4 -2 4 -2

11. A set of ordered pairs in which no two ordered pairs have the same first coordinate and no two ordered pairs have the same second coordinate. In other words: FOR EVERY X, THERE IS ONLY ONE Y. FOR EVERY Y, THERE IS ONLY ONE X. DEFINITION OF ONE-TO-ONE FUNCTION

12. f(x) = x 2 is not a one-to-one function. Thus, it has no inverse.

13. Recall a graphical test which enables us to determine whether a relation is a function. “VERTICAL LINE TEST”

14. What kind of graphical test would help us to determine whether a function was one-to- one? “HORIZONTAL LINE TEST”

15. FINDING A FORMULA FOR f -1 (x) Example: First of all, check to see if it is one-to-one. Graph it!

16. Now, find the formula: Now, find the formula: x y - 2x = 5 x y = 2x + 5

17. EXAMPLE: Find inverses for the two functions below and graph them to see symmetry. f(x) = 3x - 4 g -1 (x) does not exist.

18. RESTRICTING DOMAINS Example:f(x) = x 2 - 4x f(x) = x 2 - 4x + 4 - 4 f(x) = (x - 2) 2 - 4 Vertex: (2, - 4) Domain: (x  2)

19. CONCLUSION OF SECTION 4.2 CONCLUSION OF SECTION 4.2