1. One to One Functions A function is one to one if each y value is associated with only one x value. A one to one function passes both the vertical and horizontal line tests. One to one Not

2. The Inverse of a Function If a function f(x) is one- to- one, then its inverse f -1 (x) exists. Taking the inverse of a function exchanges the x and y values. The graphs of a function and its inverse will be symmetrical in the line y=x.

3. Example 1 Given f(x) = 2x + 3, find f -1 (x). Solution: f(x) = 2x + 3 means y = 2x + 3 Exchanging x and y gives f -1 (x): x – 3 = y 2 Therefore f -1 (x) = x – 3 2 x = 2y + 3

4. Example 2 Given find f – 1 (x). We’re not done yet… Re-write the function as an equation in x and y, and then exchange the variables. Solution:

5. We have exchanged the variables and now we must solve for y. We must state our final solution using the correct notation.

6. Example 3 Sketch the inverse of the function shown. (5,7) (0,3) (– 7,0) Consider the co- ordinates of the points shown. Exchange the x and y values. (7,5) (0, – 7) (3,0) The function and its inverse are reflections of one another over the line y = x. y = f(x) y = f – 1 (x) y = x

7. Example 4 Refer to the functions from the previous slide to complete the following information. f(x)f – 1 (x) Domain Range x intercept y intercept (5,7) (0,3) (7,5) (0, – 7) (3,0) (– 7,0)