Inverse & Composite Functions –

Inverse and composite functions are Higher-tier GCSE topics that regularly appear on AQA, Edexcel and OCR papers — typically worth three to five marks. Once you understand the underlying idea, the methods are surprisingly straightforward. An inverse function reverses what the original function does, while a composite function applies two functions one after the other. This guide explains both concepts with clear, step-by-step methods and fully worked examples to get you exam-ready.

What Are Inverse and Composite Functions?

Inverse functions

The inverse function f⁻¹(x) reverses the effect of f(x). If f takes an input and produces an output, then f⁻¹ takes that output and gives back the original input.

For example, if f(x) = 2x + 3, then f(5) = 13. The inverse function should take 13 and return 5.

The graph of f⁻¹(x) is a reflection of f(x) in the line y = x.

Composite functions

A composite function applies two functions in succession. The notation fg(x) means "apply g first, then apply f to the result".

Important: fg(x) is not the same as gf(x). The order matters.

For example, if f(x) = x + 1 and g(x) = 3x, then:

• fg(x) = f(3x) = 3x + 1
• gf(x) = g(x + 1) = 3(x + 1) = 3x + 3

Step-by-Step Method

How to find an inverse function

1. Write y = f(x), replacing f(x) with y.
2. Swap x and y (replace every y with x and every x with y).
3. Rearrange to make y the subject.
4. Write the answer as f⁻¹(x) = ...

How to find a composite function

1. Identify the inner function — the one applied first. In fg(x), g is applied first.
2. Substitute g(x) into f wherever you see x.
3. Simplify the resulting expression.

How to evaluate a composite function at a value

1. Work from the inside out. For fg(2), first find g(2), then substitute that result into f.

Worked Example 1 — Inverse Function

Question: f(x) = 4x − 7. Find f⁻¹(x).

Step 1: Write y = 4x − 7.

Step 2: Swap x and y: x = 4y − 7.

Step 3: Rearrange for y:

x + 7 = 4y

y = (x + 7) / 4

Step 4: f⁻¹(x) = (x + 7) / 4

Check: f(3) = 4(3) − 7 = 5. f⁻¹(5) = (5 + 7)/4 = 12/4 = 3 ✓

Worked Example 2 — Composite Functions

Question: f(x) = x² + 1 and g(x) = 2x − 3. Find: (a) fg(x), (b) gf(x), (c) fg(4).

(a) fg(x) = f(g(x))

Replace x in f(x) with g(x) = 2x − 3:

fg(x) = (2x − 3)² + 1

Expand: fg(x) = 4x² − 12x + 9 + 1 = 4x² − 12x + 10

(b) gf(x) = g(f(x))

Replace x in g(x) with f(x) = x² + 1:

gf(x) = 2(x² + 1) − 3 = 2x² + 2 − 3 = 2x² − 1

(c) fg(4)

First find g(4) = 2(4) − 3 = 5.

Then find f(5) = (5)² + 1 = 26.

So fg(4) = 26.

Check using the expression from (a): 4(16) − 12(4) + 10 = 64 − 48 + 10 = 26 ✓

Common Mistakes

• Applying functions in the wrong order. In fg(x), you apply g first, then f. Think of it as f(g(x)) — the innermost function runs first. Many students mistakenly apply f first.
• Forgetting to swap x and y when finding the inverse. If you just rearrange for x without swapping, you will get the wrong expression.
• Not expanding brackets fully in composites. When you substitute one function into another, expand and simplify completely. (2x − 3)² is 4x² − 12x + 9, not 4x² − 9.
• Assuming fg(x) = gf(x). Composite functions are not commutative — the order matters. Always check which function is applied first.
• Writing f⁻¹(x) as 1/f(x). The inverse function f⁻¹(x) is not the reciprocal 1/f(x). They are completely different concepts.

Exam Tips

1. Always verify your inverse. Substitute a value into f, then put the result into f⁻¹. You should get back to the original value. This is a quick and reliable check.
2. Show each substitution step. For composite functions, write out fg(x) = f(g(x)) = f(2x − 3) = ... Examiners award marks for clear method.
3. On Edexcel papers, you may be asked to solve fg(x) = k. Find the composite expression first, then set it equal to k and solve.
4. When finding the inverse of a fraction, multiply both sides by the denominator early to avoid messy algebra.

Practice Questions

Question 1: f(x) = 3x + 2. Find f⁻¹(x) and verify that f⁻¹(f(7)) = 7.

Question 2: f(x) = x + 4 and g(x) = x². Find fg(3) and gf(3).

Question 3: f(x) = (2x + 1)/5. Find f⁻¹(x).

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Related Topics

• Functions and Function Notation
• Graph Transformations
• Solving Quadratic Equations
• Algebraic Proof

Summary

• The inverse function f⁻¹(x) reverses f(x). To find it: write y = f(x), swap x and y, then rearrange for y.
• A composite function fg(x) means f(g(x)) — apply g first, then f.
• fg(x) and gf(x) are not the same — the order matters.
• The graph of f⁻¹(x) is a reflection of f(x) in the line y = x.
• Always verify your inverse by checking that f⁻¹(f(a)) = a.
• Show clear substitution steps to earn method marks in the exam.
• f⁻¹(x) is not the same as 1/f(x) — do not confuse the inverse with the reciprocal.