How to Verify Inverse Functions by Composition?

A function is like a machine that receives an input and provides the output. A function takes each input and gives exactly one output.

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: \(g\)\([\color{blue}{f(x)}\)] \(=x\)

• If both compositions return \(x\) on the relevant domains, the two functions are inverses.
• If either composition fails to return \(x\), the functions are not inverses on those domains.

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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Verifying Inverses Beyond Linear Functions

The same two-composition test works for any kind of function. You just need more careful algebra and an eye on the domain.

Example: a rational function pair

Are f(x) = (2x + 1)/(x − 3) and g(x) = (3x + 1)/(x − 2) inverses?

• f(g(x)): the top becomes 2(3x + 1)/(x − 2) + 1 = (7x)/(x − 2). The bottom becomes (3x + 1)/(x − 2) − 3 = 7/(x − 2). Dividing gives x.
• g(f(x)): the top becomes 3(2x + 1)/(x − 3) + 1 = (7x)/(x − 3). The bottom becomes (2x + 1)/(x − 3) − 2 = 7/(x − 3). Dividing gives x.

Both compositions equal x (with x ≠ 3 for f and x ≠ 2 for g), so the functions are inverses.

Example: when the domain matters

Take f(x) = x² and g(x) = √x.

• f(g(x)) = (√x)² = x for x ≥ 0.
• g(f(x)) = √(x²) = |x|. This equals x only when x ≥ 0. For x = −3, g(f(−3)) = √9 = 3, not −3.

So x² and √x are inverses only when f is restricted to x ≥ 0. This is why you should always check both directions and state the domain.

Quick Checks and Common Mistakes

Composition is the proof, but a quick check can catch errors before you start.

• Test one number. With f(x) = 2x + 5 and g(x) = (x − 5)/2, f(3) = 11 and g(11) = 3. If g did not send 11 back to 3, the functions could not be inverses.
• Look at the graph. Inverse graphs are mirror images across the line y = x. If (3, 11) is on f, then (11, 3) must be on g.

Watch for these mistakes:

• Checking only f(g(x)) and stopping there.
• Writing g(x) = 1/f(x). The reciprocal is not the inverse.
• Forgetting parentheses when you substitute, such as writing 2x − 5/2 + 5 instead of 2((x − 5)/2) + 5.
• Ignoring domain limits on square roots, logs and fractions.

Frequently Asked Questions

How do you verify inverse functions using composition?

Compute both f(g(x)) and g(f(x)) and simplify. If both equal x on the relevant domains, f and g are inverses; if either gives anything else, they are not. For f(x) = 4x − 1 and g(x) = (x + 1)/4, f(g(x)) = (x + 1) − 1 = x and g(f(x)) = (4x − 1 + 1)/4 = x, so they are inverses.

Are f(x) = ∛(5x − 4) and g(x) = (x³ + 4)/5 inverse functions?

Yes. Composing in one direction, f(g(x)) = ∛(5 · (x³ + 4)/5 − 4) = ∛(x³ + 4 − 4) = ∛(x³) = x. In the other direction, g(f(x)) = ((∛(5x − 4))³ + 4)/5 = (5x − 4 + 4)/5 = x. Both compositions return x for every real number, so the two functions are inverses.

What is the inverse composition rule?

The rule says that a function and its inverse undo each other: f(f⁻¹(x)) = x for every x in the domain of f⁻¹, and f⁻¹(f(x)) = x for every x in the domain of f. It is exactly the test used to verify inverses. For example, with f(x) = x³ and f⁻¹(x) = ∛x, both (∛x)³ and ∛(x³) simplify to x.

How do you verify inverse functions graphically?

Graph both functions on the same axes along with the line y = x. If each graph is the mirror image of the other across y = x, they are inverses. A quick check is to pick a point: if (2, 9) is on f, then (9, 2) must be on g. A graph is good evidence, but the composition test is the algebraic proof.

Is there a verifying inverse functions by composition worksheet?

Yes, these worksheets usually list pairs of functions and ask you to find f(g(x)) and g(f(x)) to decide whether each pair are inverses. Include a few non-examples, such as f(x) = x + 6 and g(x) = 6 − x, where f(g(x)) = 12 − x, so the pair fails. Also try a square-root pair to practice checking domain restrictions.

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