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 

Recommended EffortlessMath Books

For a workbook that walks through function composition and inverses with worked examples, the Algebra II for Beginners covers the full chapter. For pre-calc-level treatment with trig and exponential inverses, see the Pre-Calculus for Beginners.

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Frequently Asked Questions

How do you verify two functions are inverses?

Compute both compositions: \(f(g(x))\) and \(g(f(x))\). If both equal \(x\) on the relevant domain, the functions are inverses of each other. If either composition gives something other than \(x\), the functions are not inverses. Both directions matter — one alone is not enough.

What does f(g(x)) = x mean?

It means that whatever \(g\) does to \(x\), \(f\) undoes it, returning the original input \(x\). Example: if \(g\) adds 5 and \(f\) subtracts 5, then \(f(g(x))=(x+5)-5=x\). This is the defining property of inverse operations.

Do I need to check both compositions?

Yes. There exist function pairs where \(f(g(x))=x\) holds but \(g(f(x))\neq x\), or vice versa. This happens when domains and ranges do not match up. To prove the functions are full inverses, both compositions must reduce to \(x\) on the relevant domain.

What is the inverse of a linear function?

For \(f(x)=ax+b\) with \(a\neq 0\), the inverse is \(f^{-1}(x)=\frac{x-b}{a}\). Example: \(f(x)=3x-6\) has inverse \(f^{-1}(x)=\frac{x+6}{3}\). Check: \(f(f^{-1}(x))=3\cdot\frac{x+6}{3}-6=(x+6)-6=x\). Confirmed.

How do you find the inverse of a function?

Replace \(f(x)\) with \(y\). Swap \(x\) and \(y\). Solve the new equation for \(y\). Replace \(y\) with \(f^{-1}(x)\). Example: \(f(x)=2x+1\). \(y=2x+1 \to x=2y+1 \to y=\frac{x-1}{2}=f^{-1}(x)\).

What functions have inverses?

Only one-to-one functions have inverses — functions where each output corresponds to exactly one input. The graphical test is the horizontal-line test: if no horizontal line crosses the graph more than once, the function is one-to-one and has an inverse. \(f(x)=x^2\) fails this test on the full real line but works on \(x\geq 0\).

What is the relationship between a function and its inverse on a graph?

The graphs of \(f\) and \(f^{-1}\) are reflections of each other across the line \(y=x\). If \((a, b)\) is on the graph of \(f\), then \((b, a)\) is on the graph of \(f^{-1}\). Plotting both on the same axes makes the symmetry easy to see.

Does f^(-1) mean 1/f?

No. The notation \(f^{-1}\) means “the inverse function of \(f\),” not \(1/f\). This is the same -1 used in arc trig like \(\sin^{-1}\) (which means arcsin, not \(1/\sin\)). The reciprocal \(1/f(x)\) is written without the -1 in the function name.

Why must f(g(x)) and g(f(x)) both equal x?

Inverses must undo each other in both directions. If \(f\) sends \(a\) to \(b\), then \(g\) must send \(b\) back to \(a\). The two compositions express the round trip — out and back. If only one direction works, the functions agree in part but are not true inverses across their full domain.

What is the domain of an inverse function?

The domain of \(f^{-1}\) is the range of \(f\); the range of \(f^{-1}\) is the domain of \(f\). The inverse swaps inputs and outputs. So if \(f\) maps \([0,\infty)\) to \([0,\infty)\) via \(f(x)=\sqrt{x}\), then \(f^{-1}\) maps \([0,\infty)\) to \([0,\infty)\) via \(f^{-1}(x)=x^2\).

Related EffortlessMath Lessons

If a topic on this page feels rusty, these short lessons go deeper:

• Composition of functions
• How to find the inverse of a function
• Function notation
• Domain and range of a function
• Rules of exponents