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# Inverse Functions

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### Inverse Functions

1. 1. 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).
2. 2. 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.
3. 3. 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) = x2 x 33 33 33 f(x) y x2 9999 99 99 9 99 999 f--1(x) 33 33 x 33 3
4. 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) = x2 x 55 55 55 f(x) y x2 25 25 25 25 25 25 25 25 25 255 f--1(x) 55 55 x 55 55
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) = x2 x f(x) y 121 11 121 121 11 121 121 11 121 121 11 121 121 x2 11 121 121 121 11 121 121 f--1(x) 11 1 111 11 11 x 11 11 11 1 111
6. 6. 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)
7. 7. Graphically, the x and y values of a point are switched. If the function y = g(x) contains the points 10 8 6 x 0 1 2 3 4 y 1 2 4 8 16 2 -10 -8 -6 -4 -2 2 -2 4 6 8 10 4 then its inverse, y = g-1(x), contains the points -4 x 1 2 4 8 16 -6 y 0 1 2 3 4 -8 -10 Where is there a line of reflection?
8. 8. y = f(x) The graph of a function and its inverse are mirror images about the line y=x y=x y = f-1(x)
9. 9. 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: x = 6y − 12 x + 12 = 6y x + 12 =y 6 1 x+2= y 6
10. 10. Example 2: Given the function : y = 3x2 + 2 find the inverse: Step 1: Switch x and y: x = 3y2 + 2 Step 2: Solve for y: x = 3y 2 + 2 2 x − 2 = 3y x−2 = y2 3 x−2 =y 3