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) = x2 x 33 33 33 f(x) y x2 9999 99 99 9 99 999 f--1(x) 33 33 x 33 3
- 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 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
- 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) = 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
- 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 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?
- 9. 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)
- 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: x = 6y − 12 x + 12 = 6y x + 12 =y 6 1 x+2= y 6
- 11. 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