The document discusses inverse functions and how an inverse function undoes the operations of the original function. It provides examples of finding the inverse of functions by switching the x and y values and solving for y. The inverse of a function will be a function itself only if the original function passes the horizontal line test.

1. Inverse Functions Inverse Functions

2. 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).

3. 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.

4. 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 f --1 (x) 9 9 9 9 9 9 9 3 3 3 3 3 3 3 x 2

5. 5 5 5 5 5 5 25 25 25 25 25 25 25 25 25 25 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 f --1 (x) x 2

6. 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 f --1 (x) x 2

7. 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)

8. 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 Where is there a line of reflection? 16 8 4 2 1 y 4 3 2 1 0 x 4 3 2 1 0 y 16 8 4 2 1 x

9. The graph of a function and its inverse are mirror images about the line y = x y = f(x) y = f -1 (x) y = x

10. Find the inverse of a function : Example 1: y = 6x - 12 Step 1: Switch x and y: x = 6y - 12 Step 2: Solve for y:

11. Example 2: Given the function : y = 3x 2 + 2 find the inverse: Step 1: Switch x and y: x = 3y 2 + 2 Step 2: Solve for y:

18. Ex: Graph the function f(x)=x 2 and determine whether its inverse is a function. Graph does not pass the horizontal line test, therefore the inverse is not a function.

19. Ex: f(x)=2x 2 -4 Determine whether f -1 (x) is a function, then find the inverse equation. f -1 (x) is not a function. y = 2x 2 -4 x = 2y 2 -4 x+4 = 2y 2 OR, if you fix the tent in the basement…

20. Ex: g(x)=2x 3 Inverse is a function! y=2x 3 x=2y 3 OR, if you fix the tent in the basement…

21. Assignment