How to Verify Inverse Functions by Composition?

A function is like a machine that receives an input and provides the output. Typically, the output is a slightly changed version of the original input.

The inverses of functions may be thought of as the operations that return the input to us after we have given them the output.

In other words, inverse functions are functions that do the opposite of the actions of the function. When we need to know what the input was for a given output, these ideas come in very handy.

Related Topics

• How to Find Inverse of a Function
• How to Solve Function Notation
• How to Solve Composition of Functions

A step-by-step guide to verifying inverse functions by the composition

In this example, we will show you how to figure out if two functions are the opposite of each other in two simple steps:

• Step 1:

Plug \(g(x)\) into \(f(x)\) , then simplify: \(f\)\([\color{red}{g(x)}\)] \(=x\)

If true, move to Step 2. If false, STOP! That means \(f(x)\) and \(g(x)\) are not inverses.

• Step 2:

Plug \(f(x)\) into \(g(x)\) , then simplify: \(g\)\([\color{blue}{f(x)}\)] \(=x\)

• True again, then \(f(x)\) and \(g(x)\) are the same things. Success!
• There is no way that \(f(x)\) and \(g(x)\) are the same things.

For \(f(x)\) and \(g(x)\) to be opposites of each other, you need to show that method composition works both ways! So, when you combine function \(f\) with function \(g\), you get \(x\). It is “stylish” see how it is shown in the equation below:

Must always get \(“x”\) → \(f\)\([\color{red}{g(x)}\)] \(=x=\) \(g\)\([\color{blue}{f(x)}\)]

Verifying inverse functions by the composition – Example 1:

Are functions \(f(x)= \frac{x+3}{4}\) and \(g(x)=4x-3\) inverses?

First find \(f(g(x))\): \(f(g(x))\ =\frac{4x-3+3}{4}\)\(=\frac{4x}{4}\)\(=x\)

Then find \(g(f(x))\): \(g(f(x)) =4( \frac{x+3}{4}\))\(-3=x+3-3=x\)

So functions \(f(x)\) and \(g(x)\) are inverses because \(f(g(x))=x\) and \(g(f(x))=x\)

Verifying inverse functions by the composition – Example 2:

Are functions \(h(x)= 12x- 3\) and \(g(x)=\frac{x}{12}+3\) inverses?

First find \(h(g(x))\): \(h(g(x))=12( \frac{x}{12}+{3})-3=x+36-3=x+33\)

Then find \(g(h(x))\): \(g(h(x))=\frac{12x-3}{12}+3=\frac{4x+11}{4}\)

So functions \(h(x)\) and \(g(x)\) are not inverses because \(h(g(x))≠x\) and \(g(h(x))≠x\)

Verifying inverse functions by the composition – Example 3:

Are functions \(g(x)=\frac{x-5}{4}\) and \(f(x)=4x+5\) inverses?

First find \(g(f(x))\): \(g(f(x))=\frac{4x+5-5}{4}=\frac{4x}{4}=x\)

Then find \(f(g(x))\): \(f(g(x))=4( \frac{x-5}{4})+5=x-5+5=x\)

So functions \(g(x)\) and \(f(x)\) are inverses because \(g(f(x))=x\) and \(f(g(x))=x\)

Exercises for Verifying Inverse Functions by the Composition

Which of the functions are inverse?

1. \(\color{blue}{f(x)=\frac{1}{x}-3}\) , \(\color{blue}{g(x)=3x-1}\)
2. \(\color{blue}{f(x)=7x-20}\) , \(\color{blue}{g(x)=\frac{1}{7}x-5}\)
3. \(\color{blue}{f(x)=\frac{1}{x}-4}\), \(\color{blue}{h(x)=\frac{1}{x+4}}\)
4. \(\color{blue}{f(x)=\frac{3}{2}x-5}\), \(\color{blue}{g(x)=\frac{2}{3}x+5}\)

1. The functions are not inverses 
2. The functions are not inverses 
3. The functions are inverses 
4. The functions are not inverses