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Inverse Functions

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Inverse functions worksheet
Inverse functions worksheet

Practise finding inverse functions by reversing a rule and rearranging it. The questions progress from linear functions to cubic and square-root functions, shifted cubes and algebraic fractions at GCSE Higher level. Jump to the questions

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Worksheet preview and key skills

Worksheet preview

This eight-question Higher worksheet asks you to write every inverse as an algebraic expression in x. One question also checks that you can use the inverse you have found.

The progression begins with linear rules, then covers cubic and square-root functions before finishing with algebraic fractions.

What you’ll practise

  • reversing linear operations in the correct order;
  • isolating a cube and using a cube root to find an inverse;
  • rearranging fractional-linear functions and respecting excluded values.

Use the interactive worksheet below, or read the Topic guide for the method and worked example.

Find each inverse function. Every numbered question is worth one mark.

For fractions, use brackets around the whole numerator and denominator: (x+5)/(x-2).

Topic guide

What is an inverse function?

An inverse function undoes the original function. The notation f−1(x) means “the inverse of f”; it does not mean the reciprocal 1/f(x).

Swap and rearrange

  1. Write y = f(x).
  2. Swap x and y.
  3. Rearrange to make y the subject.
  4. Write the result as f−1(x).

For example, if f(x) = 4x − 7, swap to get x = 4y − 7. Then 4y = x + 7, so f−1(x) = (x + 7)/4.

Cubic functions

A cubic of the form f(x) = ax3 + b, where a is positive, is one-to-one. After swapping x and y, undo the addition or subtraction, divide by a, then take the cube root. For example, if f(x) = 3x3 − 2, then f−1(x) = ∛((x + 2)/3).

A shifted cubic is handled in the same way: the inverse of (x + 4)3 − 1 is ∛(x + 1) − 4.

Algebraic fractions

If f(x) = (x + 1)/(x + 3), write x = (y + 1)/(y + 3). Multiplying across gives xy + 3x = y + 1, so y(x − 1) = 1 − 3x. Therefore f−1(x) = (3x − 1)/(1 − x). The original excludes x = −3 and the inverse excludes x = 1.

Useful checks

  • Undo operations in reverse order.
  • Keep brackets around a complete numerator or denominator.
  • Check that applying the function and its inverse returns the starting value.
  • For a cubic, isolate the cubed expression before taking the cube root.