1. Warm up Graph:

2. Lesson 3-4 Inverse Functions and Relations Objective: To determine inverses of relations and functions. To graph functions and their inverses

3. One-to-One Function For y = f(x) to be a 1-1 function, each x corresponds to exactly one y, and each y corresponds to exactly one x. A 1-1 function f passes both the vertical and horizontal line tests.

4. S: Social Security function IS one-one Joe Samanth a Anna Ian Chelsea George 123456789 223456789 333456789 433456789 533456789 633456789 AmericansSSN

5. HORIZONTAL LINE TEST for a 1-1 Function The function y = f(x) is a one-to-one (1-1) function if no horizontal line intersects the graph at more than one point.

6. Do these graphs pass the horizontal line test?

8. Inverse Functions The inverse of a relation is the set of ordered pairs obtained by switching the coordinates of each ordered pair in the relation. The graph of the inverse function is the REFLECTION of the graph of the original. The inverse is denoted by

9. 123456789 223456789 333456789 433456789 533456789 633456789 SSN Joe Samantha Anna Ian Chelsea George Americans The inverse of the social security function

10. A function, f, has an inverse function, g, if and only if the function f is a one-to-one (1-1) function. Existence of an Inverse Function

11. A function, f, has an inverse function, g, if and only if f(g(x)) = x and g(f(x)) = x, for every x in domain of g and in the domain of f. Definition of an Inverse Function

12. Finding the inverse of a 1-1 function Step1: Write the equation in the form Step2: Interchange x and y. Step 3: Solve for y. Step 4: Write for y.

13. Find the inverse of Step1: Step2: Interchange x and y Step 3: Solve for y

14. = = = cont’d

15. Practice Find the inverse of: f(x)= 2x -3 f(x)= 3x-1

16. Verifying Inverse Functions To verify that 2 functions are inverse you must show that f(g(x)) = x and g(f(x)) = x

18. Practice Verify that the following functions are inverses of each other: and

19. Graphing the inverse – by hand Make a T-table of the x and y values from the original function Switch the x and y values to graph the inverse.

20. Graphing the inverse - calculator Put the original graph in Y= [2 nd ] [DRAW][8] [VARS][►][1][1][ENTER]