1. Inverse Functions Objective: To find and identify inverse functions

2. Definition We defined a function as a relationship from set A to set B. An Inverse function is a relationship from set B to set A. In other words, we switch the x and the y coordinates, and the domain and range switch.

3. Definition We defined a function as a relationship from set A to set B. An Inverse function is a relationship from set B to set A. In other words, we switch the x and the y coordinates, and the domain and range switch.

4. Definition Notice how the x and the y coordinates have switched. We can also say that the domain and range have switched.

5. Definition Notice how the x and the y coordinates have switched. We can also say that the domain and range have switched. Here, I gave you the inverse function. Now, we will look at an informal way to find the inverse function.

6. Definition of Inverse Functions Let f and g be functions such that These functions are called inverse functions since they undo each other. This can also be stated

7. Example 1 Find the inverse function of. Then verify that.

8. Example 1 Find the inverse function of. Then verify that. To find an inverse function informally, switch the x and the y and solve for y.

9. Example 1 Find the inverse function of. Then verify that. To find an inverse function informally, switch the x and the y and solve for y.

10. Example 1 Find the inverse function of. Then verify that. We will verify this fact by looking at the composition of functions.

11. Example 6 You try: Find the inverse function of.

12. Example 6 You try: Find the inverse function of.

13. Example 6 You try: Find the inverse function of.

14. Example 6 You try: Find the inverse function of.

15. Example 6 You try: Find the inverse function of.

16. Example 6 You try: Find the inverse function of.

17. Example 6 You try: Find the inverse function of.

18. Example 6 You try: Find the inverse function of.

19. Verifying Inverse Functions Which of the functions is the inverse function of ?

20. Verifying Inverse Functions Which of the functions is the inverse function of ?

21. Verifying Inverse Functions Which of the functions is the inverse function of ?

22. Verify an Inverse Function You try Show that and are inverse functions.

23. Verify an Inverse Function You try Show that and are inverse functions.

24. Graph of an Inverse Function The graph of an inverse function is the reflection of the original function on the line y = x. Along this line they are mirror images of each other.

25. Horizontal Line Test If the graph of a function passes the horizontal line test, its inverse is also a function. A function whose inverse is also a function is said to be a one-to-one function.

26. Homework Pages 248-249 1-7 odd 9-15 25-31 odd