> ## Content Index
> Fetch the complete content index at: https://www.esheets.io/llms.txt
> Use this file to discover other available public pages before exploring further.

# Inverse Functions
- URL: https://www.esheets.io/inverse-functions/
- Published: 2026-10-17T19:00:32.000Z
- Updated: 2026-10-17T19:07:30.000Z
- Author: Richard Linnington
- Tags: Maths, Algebra

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](#practise-now)

## Practise now

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](#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*) = 4*x* − 7, swap to get *x* \= 4*y* − 7\. Then 4*y* \= *x* \+ 7, so *f*−1(*x*) = (*x* \+ 7)/4.

### Cubic functions

A cubic of the form *f*(*x*) = *a* *x*3 \+ *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*) = 3*x*3 − 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* \+ 3*x* \= *y* \+ 1, so *y*(*x* − 1) = 1 − 3*x*. Therefore *f*−1(*x*) = (3*x* − 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.