1. Warm Up

2. Objective: To find the inverse of a function, if the inverse exists.

4. Functions Imagine functions are like the dye you use to color eggs. The white egg (x) is put in the function blue dye, B(x), and the result is a blue egg (y).

5. The Inverse Function “undoes” what the function does. The Inverse Function of the Blue dye is bleach. The bleach will “undye” the blue egg and make it white.

6. In the same way, the inverse of a given function will “undo” what the original function did. For example, let’s take a look at the square function: f(x) = x 2 3 x f(x) 3 3 3 3 3 9 9 9 9 9 9 9 y 9 9 9 9 9 99 3 3 3 3 3 3 3 x2x2

7. 5 5 5 5 5 5 25252525 25252525 25252525 25252525 25 25252525 25252525 25252525 25252525 25252525 5 5 5 5 5 5 5 5 5 In the same way, the inverse of a given function will “undo” what the original function did. For example, let’s take a look at the square function: f(x) = x 2 x f(x) y x2x2

8. 11 11 11 11 11 11 121 121 121 121 121 121 121 121 121 121 121 121 121 121 11 11 11 11 11 11 11 11 In the same way, the inverse of a given function will “undo” what the original function did. For example, let’s take a look at the square function: f(x) = x 2 x f(x) y x2x2

9. Inverse Function Definition Two functions f and g are called inverse functions if the following two statements are true:

10. Graphically, the x and y values of a point are switched. The point (4, 7) has an inverse point of (7, 4) AND The point (-5, 3) has an inverse point of (3, -5)

11. Graphically, the x and y values of a point are switched. If the function y = g(x) contains the points then its inverse, y = g -1 (x), contains the points x01234 y124816 x1248 y01234 Where is there a line of reflection?

12. The graph of a function and its inverse are mirror images about the line y = x

13. Inverse Notes

14. Find the inverse of a function algebraically: Example 1: f(x) = 6x - 12 Example 1: f(x) = 6x - 12 Step 1: Switch x and y x = 6y - 12 Step 2: Solve for y *Note: You can replace f(x) with y.

15. Given the function: f(x) = 3x 2 + 2 Find the inverse. Step 1: Switch x and y x = 3y 2 + 2 Step 2: Solve for y Example 2:

16. On the same axes, sketch the graph of and its inverse. Notice x Solution:

17. On the same axes, sketch the graph of and its inverse. Notice Solution: Using the translation of what is the equation of the inverse function?

18. domain and range The domain of is. Since is found by swapping x and y, Domain Range The previous example used. the values of the domain of give the values of the range of.

19. domain and range The previous example used. The domain of is. Since is found by swapping x and y, give the values of the domain of the values of the domain of give the values of the range of. Similarly, the values of the range of

20. SUMMARY The graph of is the reflection of in the line y = x. At every point, the x and y coordinates of become the y and x coordinates of. The values of the domain and range of swap to become the values of the range and domain of.

21. Example 3 Consider the functions f and g listed below. Show that f and g are inverses of each other. a. show graphically b. show with a table c. show algebraically

22. a. graphically b. using table of values

23. Solution to example 3 Algebraically

24. Vertical and Horizontal Line Test Does the graph pass the vertical line test? Does the graph pass the horizontal line test? What does passing/not passing the horizontal line test mean?

25. The Horizontal-Line Test One-to-One Function A function for which every element of the range corresponds to exactly one element of the domain.

26. Example 4 Restricted Domain A.) Graph y = f -1 (x) B.) Find a rule for f -1 (x)

27. 4.5 Inverse Functions Visualize the Inverse Root Function?

28. Example 4 on your calculator

30. Homework Page 149 #1-27 odd, 30

31. Quiz 4.3 – 4.5 4.3 Reflections Symmetry 4.4 Period & Amplitude Stretching & Translating Graphs 4.5Find Inverse Function