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To find an inverse function, write it as y = f(x), rearrange to make x the subject, then swap x and y. Free IGCSE Maths notes and 8 FAQs.
Composite and Inverse Functions is part of Functions, Coordinates and Linear Graphs in IGCSE Edexcel Higher Maths.
2 min
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20
Lessons in this topic
IGCSE Edexcel Higher
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Reviewed by Revise Right
Content review
To find an inverse function, write it as y = f(x), rearrange to make x the subject, then swap x and y. Free IGCSE Maths notes and 8 FAQs.
Use this lesson to stay inside Functions, Coordinates and Linear Graphs while connecting the detail back to the wider Topic 3: Sequences, Functions and Graphs topic.
Read the core explanations from Composite and Inverse Functions before testing yourself from memory.
Composite functions are about order. Doing g first and then f is not the same as doing f first and then g unless the rules happen to match.
Inverse functions are about reversing a process. If one function does and then undoing all of that gets you back where you started, the two are inverses.
Worked example: Evaluating a composite function
Let f(x) = 2x + 1 and g(x) = x - 4. Find f(g(6)) and f(g(x)).
Answer: f(g(6)) = 5 and f(g(x)) = 2x - 7
Worked example: Finding an inverse
Find the inverse of f(x) = 3x + 2.
Answer: f^-1(x) = (x - 2)/3
When you see f(g(x)), resist the urge to expand immediately. Write the inner result first so the order is explicit.
Worked example: Checking an Inverse by Composition
For f(x) = 3x + 2 and f^-1(x) = (x - 2)/3, show that f(f^-1(x)) = x.
Answer: f(f^-1(x)) = x, so the inverse is correct.
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-25.
Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
Write the function as y = ..., then rearrange to make x the subject and replace y with x. For f(x) = 3x + 2, y = 3x + 2 gives x = (y - 2)/3, so f⁻¹(x) = (x - 2)/3.
With f(x) = 2x + 1 and g(x) = x - 4, f(g(6)) = f(2) = 5, because g(6) = 2 is worked out first. Algebraically, f(g(x)) = 2(x - 4) + 1 = 2x - 7.
For f(g(x)), work inside out: evaluate g(x) first, then substitute that result into f. With f(x) = 2x + 1 and g(x) = x - 4, g(6) = 2 so f(g(6)) = 5, and f(g(x)) = 2(x - 4) + 1 = 2x - 7.
An inverse function is checked by composition: substituting it into the original function should give x back. For f(x) = 3x + 2 and f⁻¹(x) = (x - 2)/3, f(f⁻¹(x)) = 3((x - 2)/3) + 2 = x - 2 + 2 = x.
f(g(x)) is not usually the same as g(f(x)). Composite functions are about order: doing g first and then f is not the same as doing f first and then g unless the rules happen to match.
The inverse of f(x) = 3x + 2 is f⁻¹(x) = (x - 2)/3. Writing y = 3x + 2 and rearranging gives x = (y - 2)/3, then y is replaced with x in inverse notation.
f⁻¹(x) is the inverse function of f: a function that reverses the effect of f. Applying f and then f⁻¹ gets back to where it started.
An inverse is found by undoing each operation in the opposite order. If the rule is multiply by 3 then add 5, the inverse is subtract 5 then divide by 3.