Inverse functions GCSE Questions and Worked Answers

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Start with the meaning

What you need to know about inverse functions

Imagine a number machine: it multiplies your number by 3 and then adds 2. If you put in 4, you get 14. To recover the original 4 from 14, first subtract 2, then divide by 3. A rule that takes the result back to the starting number is called an inverse function.

See the idea first

The return journey reverses the steps

Name the forward rule f. Writing f(4) = 14 means its input is 4 and its output is 14. Writing f⁻¹(14) = 4 means the inverse takes 14 back to 4. The small −1 is a label for the inverse, not an instruction to take a reciprocal.

A rule and its undoing
Original input	f: multiply by 3, then add 2	Inverse: subtract 2, then divide by 3
1	5	1
4	14	4
7	23	7

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Recover one input

What the problem asks: f(x) = 5x − 4. Find f⁻¹(21).

How to solve it: 21 is the output. Add 4 to get 25, then divide by 5 to get the original input 5.

Find the inverse rule

What the problem asks: Find f⁻¹(x) when f(x) = 5x − 4.

How to solve it: Write y = 5x − 4. Rearrange to x = (y + 4)/5. The inverse takes y as its input; rename that input x to write f⁻¹(x) = (x + 4)/5.

Check whether undoing is unique

What the problem asks: Can y = x² be undone uniquely if any real input is allowed?

How to solve it: No: both 3 and −3 produce 9. Restricting the original inputs to x ≥ 0 makes the square-root rule a unique inverse.

A reliable routine

Find an inverse formula

Use this when a one-to-one rule must be undone. One-to-one means different allowed inputs give different outputs, so each output can return to exactly one input.

Write y = the forward rule.
Rearrange until x is alone, undoing the last operation first.
Rename the inverse input y as x and label the result f⁻¹(x).
Check by taking a number forwards and then backwards; keep any input restrictions.

Check: An inverse is a whole return rule. For f(x) = 2x, the inverse is x/2, whereas 1/f(x) is 1/(2x).

Fully worked

Inverse functions GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Undo a numeric output

Higher only2 marks

Question

f(x) = 4x + 7. Find $f^{-1}(31)$ .

Set the forward output to 31.

4x+7=31

4x=24

x=6

Therefore $f^{-1}(31)=6$ . Check: 4 × 6 + 7 = 31.

Example 2

A linear inverse

Higher only3 marks

Question

f(x) = 3x − 8. Find $f^{-1}(x)$ .

y=3x-8

y+8=3x

x=\frac{y+8}{3}

Rename the inverse input:

f^{-1}(x)=\frac{x+8}{3}

Add 8 before dividing by 3.

Example 3

A fraction rule

Higher only3 marks

Question

g(x) = (x + 5)/4. Find $g^{-1}(x)$ .

y=\frac{x+5}{4}

4y=x+5

x=4y-5

Thus

g^{-1}(x)=4x-5

Multiplication by 4 undoes the final division.

Example 4

A negative coefficient

Higher only3 marks

Question

h(x) = 9 − 2x. Find $h^{-1}(x)$ and $h^{-1}(3)$ .

y=9-2x

2x=9-y

x=\frac{9-y}{2}

Hence $h^{-1}(x)=(9-x)/2$ and

h^{-1}(3)=\frac{9-3}{2}=3

Check: h(3) = 3. Here 3 happens to be sent to itself, so going back also gives 3.

Example 5

A restricted quadratic

Higher only4 marks

Question

f(x) = x² + 6 for x ≥ 0. Find its inverse and state the allowed inputs of the inverse.

y=x^2+6

x^2=y-6

x=\sqrt{y-6}

Choose the non-negative root because the original x is non-negative.

f^{-1}(x)=\sqrt{x-6},\quad x\ge6

The original outputs are at least 6, so these are the inverse's allowed inputs.

Example 6

Verify a proposed inverse

Higher only3 marks

Question

f(x) = 2x − 3. A student says $f^{-1}(x)=x/2+3$ . Correct the rule and test it using input 4 in f.

The last forward step is subtracting 3, so undo it before dividing.

f^{-1}(x)=\frac{x+3}{2}

Forward: f(4) = 5. Backward:

f^{-1}(5)=\frac{5+3}{2}=4

The proposed rule gives 5.5, not 4.

10 original questions · total 27 marks

Inverse functions GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 32 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed

1

Recover a number

Higher only2 marks

f(x) = 6x + 1. Find $f^{-1}(25)$ .

Show worked answer

6x+1=25

6x=24

x=4

2

Subtract then multiply

Higher only2 marks

f(x) = 2(x − 5). Find $f^{-1}(14)$ .

Show worked answer

Undo the doubling first, then undo the subtraction: halve 14 to get 7, then add 5 to get 12.

14\div2+5=7+5=12

Check f(12) = 14.

3

Linear rule

Higher only3 marks

Find the inverse of f(x) = 7x + 4.

Show worked answer

y=7x+4

x=\frac{y-4}{7}

Therefore $f^{-1}(x)=(x-4)/7$ .

4

Division rule

Higher only3 marks

Find the inverse of g(x) = x/3 − 2.

Show worked answer

y+2=x/3

x=3(y+2)

Thus $g^{-1}(x)=3(x+2)$ . The whole bracket is multiplied by 3.

5

Negative slope

Higher only3 marks

Find the inverse of h(x) = 5 − 4x.

Show worked answer

4x=5-y

x=(5-y)/4

Thus $h^{-1}(x)=(5-x)/4$ .

6

Two-step fraction

Higher only3 marks

Find the inverse of f(x) = (2x − 1)/5.

Show worked answer

5y=2x-1

2x=5y+1

f^{-1}(x)=\frac{5x+1}{2}

7

Cube rule

Higher only3 marks

Find the inverse of f(x) = x³ + 2 for all real x.

Show worked answer

y-2=x^3

x=\sqrt[3]{y-2}

Thus $f^{-1}(x)=\sqrt[3]{x-2}$ . Every real number, including a negative number, has exactly one real cube root, so no restriction is needed.

8

Restricted square

Higher only4 marks

f(x) = x² − 4 for x ≥ 0. Find its inverse and its allowed inputs.

Show worked answer

x^2=y+4

x=\sqrt{y+4}

Thus $f^{-1}(x)=\sqrt{x+4}$ for x ≥ −4. Use the non-negative root to match the original restriction.

9

Inverse is not reciprocal

Higher only2 marks

For f(x) = 3x, compare $f^{-1}(6)$ with $1/f(6)$ .

Show worked answer

f^{-1}(6)=6/3=2

1/f(6)=1/18

They answer different questions: undo the rule, or take the reciprocal of its output.

10

Explain a restriction

Higher only2 marks

Why does f(x) = x² fail to have an inverse function on all real inputs?

Show worked answer

Two different inputs can give the same output: f(2) = f(−2) = 4. A return rule could not choose one original input for 4. Restricting the original inputs, for example to x ≥ 0, removes that ambiguity.

Examiner-style feedback

Common inverse functions mistakes

Undoing in forward order

For 3x + 8, subtract 8 first, then divide by 3.

Reciprocal notation

f⁻¹(x) names an inverse function, not 1/f(x).

Dropping restrictions

The allowed outputs of the original rule become the allowed inputs of its inverse.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

An inverse recovers the original input.
Undo in reverse order.
Rearrange before renaming the input.
Check uniqueness and restrictions.

Quick answers

Inverse functions FAQ

Must I swap x and y first?

You can, but you do not have to. Rearranging y = f(x) for x and then naming the inverse input is equivalent.

Does every function have an inverse?

Not on its full set of inputs. Each output must identify one original input; sometimes a restriction is needed.

Build connected skills

What to revise next

Related skill

Composite functions

Combine a rule with its inverse.

Revise composite functions

Related skill

Rearranging formulae

Practise undoing algebraic operations.

Revise rearranging formulae

Content standards

Curriculum and rights review

A7 Higher-tier content. All definitions and worked tasks are introduced from simple inputs. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references

AQA GCSE Maths subject content