Lesson Objectives
• Learn how to determine if two functions are inverses

## How to Determine if Two Functions are Inverses

How can we determine if two functions are inverses of each other? We previously stated that the domain of a function becomes the range of its inverse and the range of a function becomes the domain of its inverse. Let's take a simple function such as: $$f(x)=2x$$ This function maps an x-value of 5 to a y-value of 10. $$f(5)=2(5)=10$$ If we find the inverse of the function, it will map an x-value of 10 to a y-value of 5. $$f^{-1}(x)=\frac{x}{2}$$ $$f^{-1}(10)=\frac{10}{2}=5$$ We can think about the inverse as reversing the procedure of the original function. In the original function, we multiply 2 by an unknown number (x). In the inverse, we take the unknown number (x) and divide by 2. We know that multiplying by 2 and dividing by 2 are opposite operations and will cancel each other out. What happens if we plug in f-1(x) in for x in the original function? $$f(f^{-1}(x))=2\left(\frac{x}{2}\right)$$ $$\require{cancel}f(f^{-1}(x))=\cancel{2}\left(\frac{x}{\cancel{2}}\right)=x$$ Based on our example, we can state the following rule:
For inverses f and f-1: $$f(f^{-1}(x))=x$$ $$f^{-1}(f(x))=x$$ In other words, we can prove that two functions are inverses by plugging the inverse function in for x in the original function. If they are inverses, the results should just be x. We also need to check the other scenario. We will plug in the original function in for x in the inverse function. Again, if they are inverses, the result should just be x.
In other words, if f(x) and g(x) are inverses:
f(g(x)) = x
g(f(x)) = x
Let's look at an example.
Example 1: Determine if the functions f and g are inverses. $$f(x)=\sqrt[5]{x + 1}$$ $$g(x)=x^5 - 1$$ To determine if these two functions are inverses, let's plug g(x) in for x in f(x). $$f(g(x))=\sqrt[5]{(x^5 - 1) + 1}$$ $$f(g(x))=\sqrt[5]{x^5}$$ $$f(g(x))=x$$ Now we will plug f(x) in for x in g(x). $$g(f(x))=(\sqrt[5]{x + 1})^5 - 1$$ $$g(f(x))=x + 1 - 1$$ $$g(f(x))=x$$ In each case, we get x as a result. We can state that these two functions are inverses.

#### Skills Check:

Example #1

Determine if the two functions are inverses. $$f(x)=\sqrt[5]{x}- 2$$ $$g(x)=(x + 2)^5$$

A
Inverses
B
Not Inverses

Example #2

Determine if the two functions are inverses. $$f(x)=\frac{2}{3}x + \frac{2}{3}$$ $$g(x)=5x - 5$$

A
Inverses
B
Not Inverses

Example #3

Determine if the two functions are inverses. $$g(x)=\frac{2}{x - 1}+ 1$$ $$f(x)=\frac{2}{x - 1}+ 1$$