Sheet № 30 · Higher only · AQA · Edexcel · OCR
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
§Key definitions
Important:
fg(x) is not the same as gf(x). The order matters.
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:
§Formulas to memorise
If f(a) = b, then f⁻¹(b) = a
fg(x) = f(g(x))
fg(x) = f(3x) = 3x + 1
gf(x) = g(x + 1) = 3(x + 1) = 3x + 3
Important:: fg(x) is not the same as gf(x). The order matters.
Write — y = f(x), replacing f(x) with y.
Swap — x and y (replace every y with x and every x with y).
Rearrange — to make y the subject.
Write — the answer as f⁻¹(x) = ...
Identify the inner function — the one applied first. In fg(x), g is applied first.
Worked example
f(x) = 4x − 7. Find f⁻¹(x).
Step 1: Write y = 4x − 7.
⚠ 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
- →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.
- →Show each substitution step. For composite functions, write out fg(x) = f(g(x)) = f(2x − 3) = ... Examiners award marks for clear method.
- →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.
- →When finding the inverse of a fraction, multiply both sides by the denominator early to avoid messy algebra.