1. Inverse Functions

2. Definition of the Inverse Function
Let f and g be two functions such that
f (g(x)) = x for every x in the domain of g
and g(f (x)) = x for every x in the domain of f.
The function g is the inverse of the function f, and denoted by f -1 (read “f-inverse”).
Thus, f ( f -1(x)) = x and f -1( f (x)) = x.
The domain of f is equal to the range of f -1, and vice versa.

3. Text Example Show that each function is the inverse of the other:
f (x) = 5x and g(x) = x/5.
Solution
To show that f and g are inverses of each other, we must show that
f (g(x)) = x and g( f (x)) = x.
f (g(x)) = 5g(x) = 5(x/5) = x.
g(f (x)) = f (x)/5 = 5x/5 = x.
Notice how f -1 undoes the change produced by f.

4. Finding the Inverse of a Function
The equation for the inverse of a function f can be found as follows:
Replace f (x) by y in the equation for f (x).
Interchange x and y.
Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function.
If f has an inverse function, replace y in step 3 with f -1(x). We can verify our result by showing that f ( f -1(x)) = x and f -1( f (x)) = x.

5. Text Example Find the inverse of f (x) = 7x – 5. Solution
Step 1 Replace f (x) by y.
y = 7x – 5
Step Interchange x and y.
x = 7y – 5
This is the inverse function.
Step Solve for y.
x + 5 = 7y
Add 5 to both sides.
x + 5 = y 7
Divide both sides by 7.
Step Replace y by f -1(x).
x + 5 7
f -1(x) =
Rename the function f -1(x).

6. One-to-One Functions
If f is a one-to-one function then its inverse will also be a function.
A function is one-to-one if for different values of x there are different values of f(x).
The easiest way to test for this property is to use the horizontal line test.

7. The Horizontal Line Test For Inverse Functions
A function f has an inverse that is a function, f –1, if there is no horizontal line that intersects the graph of the function f at more than one point.

8. Example
Does f(x) = x2+3x-1 have an inverse function?

9. Example
Solution:
This graph does not pass the horizontal line test, so f(x) = x2+3x-1 does not have an inverse function.

10. Example
The graph an inverse function is the reflection of the original function about the line y = x.
Graph the following functions and their inverse functions on the same axes: