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Inverse Functions

HigherHigher tier onlyAQAEdexcelOCR

This free Higher GCSE Maths worksheet on inverse functions helps you revise finding inverse functions. Questions build from straightforward to exam standard, with full worked answers in the mark scheme — ideal for revision or homework. Suitable for AQA, Edexcel and OCR. Swap x and y and rearrange to find the inverse.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

An inverse function undoes what the original function does. If \(f\) turns \(3\) into \(11\), then \(f^{-1}\) turns \(11\) back into \(3\). The notation is \(f^{-1}(x)\).

Finding it is the same process as changing the subject of a formula. Write \(y = f(x)\), rearrange to make \(x\) the subject, then swap the letters so the answer is in terms of \(x\).

The superscript \(-1\) here does not mean a reciprocal. \(f^{-1}(x)\) is the inverse function, not \(\frac{1}{f(x)}\), and confusing the two leads to completely wrong answers.

Revision notes

Finding an inverse

Set \(y\) equal to the function, rearrange for \(x\), then replace \(y\) with \(x\).

For \(f(x) = 3x + 5\): write \(y = 3x + 5\), rearrange to \(x = \frac{y-5}{3}\), so \(f^{-1}(x) = \frac{x-5}{3}\).

Checking the inverse

Applying a function then its inverse returns the original value.

So \(f^{-1}f(x) = x\). Testing with a number is quick: \(f(2) = 11\) and \(f^{-1}(11) = 2\), confirming the inverse is right.

Not a reciprocal

The notation \(f^{-1}\) means the inverse function, not one over the function.

For \(f(x) = 3x + 5\), the inverse is \(\frac{x-5}{3}\), not \(\frac{1}{3x+5}\).

Key points

  • An inverse function undoes the original.
  • Write \(y = f(x)\) and make \(x\) the subject.
  • Swap the letters at the end.
  • \(f^{-1}f(x) = x\).
  • \(f^{-1}\) does not mean a reciprocal.
  • Check by substituting a number both ways.

Worked examples

Example 1

Find the inverse of \(f(x) = 2x - 7\).

Working

\[y = 2x - 7\]write y equal to the function
\[x = \frac{y + 7}{2}\]rearrange to make x the subject
\[f^{-1}(x) = \frac{x + 7}{2}\]swap the letters

Example 2

Find the inverse of \(f(x) = \dfrac{x}{4} + 1\).

Working

\[y = \frac{x}{4} + 1\]write y equal to the function
\[x = 4(y - 1)\]subtract 1 then multiply by 4
\[f^{-1}(x) = 4(x - 1)\]swap the letters

Example 3

Check that the inverse of \(f(x) = 5x\) is \(f^{-1}(x) = \dfrac{x}{5}\).

Working

\[f(3) = 15\]apply the function to a test value
\[f^{-1}(15) = 3\]apply the inverse to the result
\[\text{Returns the original, so correct}\]the inverse undoes the function

Common mistakes

  • Reading \(f^{-1}\) as a reciprocal.

    It means the inverse function, not one divided by the function.

  • Forgetting to swap the letters.

    The final answer must be written in terms of x.

  • Rearranging incorrectly.

    Undo the operations in reverse order, as with changing the subject.

  • Not checking the answer.

    Applying f then f⁻¹ to a number should return the original.

Exam tips

  • Treat it exactly as a change-of-subject problem.
  • Swap the letters as the final step, not before.
  • Test with a number to confirm the inverse works.
  • Remember the −1 is notation, not a power.

Key terms

Inverse function
A function that reverses another.
Subject
The letter isolated on one side of a formula.
Notation
The symbols used to express a mathematical idea.
Reciprocal
One divided by a value, which an inverse function is not.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.