GCSE Maths · Algebra

Functions GCSE Questions, Worked Examples and Answers

A function is a rule that gives one output for each permitted input. Substitute carefully for ordinary function values, work from the inside out for composite functions, and reverse the operations in reverse order for an inverse function.

Edexcel · AQA · OCRFoundation basics · Higher inverse & composite15 original questions

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Follow the input

What you need to know about GCSE functions

A function is a rule that takes an input and produces one output. For the rule ‘multiply by 3, then subtract 2’, an input of 5 gives 13. If we name this rule f, the statement f(5) = 13 records the same input-and-output journey; f(x) does not mean f multiplied by x.

Input → rule → output

Read a function machine from input to output

The machine x → ×3 → −2 → y represents y = 3x − 2. For input 5, follow the boxes from left to right.

Input5replace x by 5

Apply rule3 × 5 − 2keep the order of operations

Output13so f(5) = 13

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.

Evaluate a function

What the problem asks: You are given a rule such as f(x) and a specific input such as f(−3).

How to solve it: Substitute the input wherever x appears, using brackets for negative or fractional values.

$f(a)=\text{the rule with }x\text{ replaced by }a$

Find an input

What the problem asks: The output is given, for example f(x) = 17, and x is unknown.

How to solve it: Set the function rule equal to the output and solve the resulting equation.

Form a function

What the problem asks: A context describes a fixed amount and a changing amount per item.

How to solve it: Use the input as a variable and translate each operation in the order described.

Evaluate a composite function

What the problem asks: Two function names appear together, such as fg(3) or f(g(3)).

How to solve it: Work from the inside out: apply g first in fg(x), then put that output into f.

$fg(x)=f(g(x))$

Find a composite rule

What the problem asks: The question asks for fg(x) in terms of x.

How to solve it: Substitute the complete inner rule into every x of the outer rule, using brackets.

Apply one function twice

What the problem asks: The same function name is repeated, such as ff(2) or f(f(x)).

How to solve it: Use the first output as the second input. Work from the function nearest the input, just as for any composite.

Find an inverse function

What the problem asks: The notation f⁻¹(x) asks for a rule that reverses f.

How to solve it: Write y = f(x), swap x and y, then rearrange for y. This reverses the input-output mapping.

$f^{-1}(f(x))=x$

A reliable routine

A routine for evaluating or composing functions

Use this routine when a question asks you to apply one or more named function rules. Translate the notation into an input-and-output journey before calculating so the order stays visible. Inverse-function problems use the separate reversing method above.

Identify the input, the output and the exact function rule being applied.
Rewrite a composite as nested brackets: fg(x) means f(g(x)).
Substitute the entire input expression using brackets, then simplify.

Check: Substitute a simple value into your final rule. A valid inverse should take an output of f back to the original input, and fg will usually differ from gf.

Fully worked

Functions GCSE worked examples

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

Example 1

Evaluate a linear function

2 marks

Question

Given $f(x)=3x-4$ , find $f(6)$ .

Recognise it: $f(6)$ gives the input directly.

Why this method: substitute $6$ for $x$ in the rule.

$f(6)=3(6)-4=18-4=\boxed{14}$

Example 2

Use a negative input

2 marks

Question

Given $g(x)=x^2+2x$ , find $g(-3)$ .

Recognise it: evaluate the rule at a negative input.

Why this method: brackets preserve the negative value when it is squared.

$g(-3)=(-3)^2+2(-3)=9-6=\boxed{3}$

Example 3

Find the input from an output

3 marks

Question

Given $f(x)=5x+7$ , solve $f(x)=42$ .

Recognise it: the output $42$ is known and the input $x$ is missing.

Why this method: set the rule equal to the stated output and solve.

$5x+7=42$

$5x=35$

$\boxed{x=7}$

Example 4

Build a function from a context

3 marks

Question

A taxi fare is £4.50 plus £2.80 for each mile travelled. Write a function $C(m)$ for the cost in pounds of a journey of $m$ miles.

Recognise it: £4.50 is fixed and £2.80 changes with the number of miles.

Why this method: multiply the per-mile charge by $m$ , then add the fixed fare.

$\boxed{C(m)=2.80m+4.50}$

Writing $2.80$ and $4.50$ keeps the amounts in pounds and pence.

Example 5

Compare fg and gf

Higher only4 marks

Question

Given $f(x)=2x+1$ and $g(x)=x^2$ , find $fg(3)$ and $gf(3)$ .

Recognise it: these are composite functions and the order changes.

Why this method: work from the function nearest the input. In $fg(3)$ , apply $g$ first.

$fg(3)=f(g(3))=f(9)=2(9)+1=19$

For $gf(3)$ , apply $f$ first.

$gf(3)=g(f(3))=g(7)=7^2=49$

$\boxed{fg(3)=19,\quad gf(3)=49}$

Example 6

Form a composite function

Higher only3 marks

Question

Given $f(x)=3x-2$ and $g(x)=x+5$ , find $fg(x)$ in its simplest form.

Recognise it: $fg(x)$ means $f(g(x))$ .

Why this method: substitute the whole inner rule $x+5$ into f.

$fg(x)=3(x+5)-2$

$=3x+15-2$

$\boxed{fg(x)=3x+13}$

Example 7

Find an inverse function

Higher only4 marks

Question

Given $f(x)=4x-9$ , find $f^{-1}(x)$ .

Recognise it: $f^{-1}$ must reverse the original input-output rule.

Why this method: the original machine multiplies by $4$ , then subtracts $9$ . The inverse machine undoes those operations in reverse order: add $9$ , then divide by $4$ .

$x\ \xrightarrow{\times4}\ 4x\ \xrightarrow{-9}\ f(x)$

$x\ \xrightarrow{+9}\ x+9\ \xrightarrow{\div4}\ \frac{x+9}{4}$

So $f^{-1}(x)=\frac{x+9}{4}$ .

The algebraic method gives the same result. Write the output as $y$ , swap input and output, then rearrange.

$y=4x-9$

Swap $x$ and $y$ :

$x=4y-9$

$x+9=4y\quad\Rightarrow\quad y=\frac{x+9}{4}$

$\boxed{f^{-1}(x)=\frac{x+9}{4}}$

Check using input $5$ :

$f(5)=11,\qquad f^{-1}(11)=\frac{11+9}{4}=5\ \checkmark$

Example 8

Solve a composite equation

Higher only5 marks

Question

Given $f(x)=2x-1$ and $g(x)=x^2+3$ , solve $gf(x)=84$ .

Recognise it: $gf(x)=g(f(x))$ , so the linear output goes into the quadratic rule.

Why this method: form the composite first, then solve the quadratic equation.

$gf(x)=(2x-1)^2+3$

$(2x-1)^2+3=84$

$(2x-1)^2=81$

$2x-1=9\quad\text{or}\quad2x-1=-9$

$\boxed{x=5\quad\text{or}\quad x=-4}$

15 original questions · total 45 marks

Functions GCSE exam-style questions

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

Before you startAllow about 50 minutes · show substitutions and keep the function order visible · answers start collapsed

Substitute an input

1 mark

Given $f(x)=4x+3$ , find $f(5)$ .

Show worked answer

Recognise it: replace $x$ by $5$ .

$f(5)=4(5)+3=\boxed{23}$

Substitute a negative input

2 marks

Given $f(x)=x^2-5$ , find $f(-4)$ .

Show worked answer

Recognise it: brackets are needed around the negative input.

$f(-4)=(-4)^2-5=16-5=\boxed{11}$

Evaluate a fractional rule

2 marks

Given $h(x)=\frac{x}{3}+7$ , find $h(12)$ .

Show worked answer

Recognise it: substitute the given input into the complete rule.

$h(12)=\frac{12}{3}+7=4+7=\boxed{11}$

Solve for the input

2 marks

Given $f(x)=7x-2$ , solve $f(x)=40$ .

Show worked answer

Recognise it: set the rule equal to the known output.

$7x-2=40$

$7x=42\quad\Rightarrow\quad\boxed{x=6}$

Use a cost function

3 marks

A phone plan costs £8 per month plus £0.06 per text. Write a function $C(t)$ for the monthly cost of $t$ texts, then find $C(120)$ .

Show worked answer

Recognise it: £8 is fixed and £0.06 is paid for each text.

$C(t)=8+0.06t$

$C(120)=8+0.06(120)=8+7.2$

$\boxed{C(t)=8+0.06t,\quad C(120)=\text{£}15.20}$

Complete function outputs

3 marks

For $f(x)=2x^2-1$ , find the outputs when $x=-2$ , $x=0$ and $x=3$ .

Show worked answer

Recognise it: evaluate the same rule at three separate inputs.

$f(-2)=2(-2)^2-1=7$

$f(0)=2(0)^2-1=-1$

$f(3)=2(3)^2-1=17$

$\boxed{7,\ -1,\ 17}$

Evaluate in both orders

Higher only3 marks

Given $f(x)=x+4$ and $g(x)=3x$ , find $fg(2)$ and $gf(2)$ .

Show worked answer

Recognise it: work from the inside function each time.

$fg(2)=f(g(2))=f(6)=10$

$gf(2)=g(f(2))=g(6)=18$

$\boxed{fg(2)=10,\quad gf(2)=18}$

Simplify a composite rule

Higher only3 marks

Given $f(x)=2x+5$ and $g(x)=x-3$ , find $fg(x)$ .

Show worked answer

Recognise it: $g$ is the inner function.

$fg(x)=f(x-3)=2(x-3)+5$

$=2x-6+5=\boxed{2x-1}$

Apply the same function twice

Higher only3 marks

Given $f(x)=3x-2$ , find $ff(4)$ and write $ff(x)$ in its simplest form.

Show worked answer

Recognise it: $ff$ means $f(f(x))$ , so use each output as the next input.

$f(4)=3(4)-2=10$

$ff(4)=f(10)=3(10)-2=\boxed{28}$

For a general input, substitute the whole expression $3x-2$ into $f$ .

$ff(x)=3(3x-2)-2$

$\boxed{ff(x)=9x-8}$

Find a linear inverse

Higher only3 marks

Given $f(x)=3x+8$ , find $f^{-1}(x)$ .

Show worked answer

Recognise it: reverse the input-output relationship.

$y=3x+8$

Swap: $x=3y+8$ .

$y=\frac{x-8}{3}$

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

Solve when two composite functions are equal

Higher only5 marks

Given $f(x)=2x+1$ and $g(x)=x^2$ , solve $fg(x)=gf(x)$ .

Show worked answer

Recognise it: form each composite in the stated order, then solve the resulting equation.

$fg(x)=f(x^2)=2x^2+1$

$gf(x)=g(2x+1)=(2x+1)^2$

Set the expressions equal.

$2x^2+1=4x^2+4x+1$

$0=2x^2+4x$

$0=2x(x+2)$

$\boxed{x=0\quad\text{or}\quad x=-2}$

Evaluate an inverse

Higher only2 marks

Given $f(x)=4x-7$ , find $f^{-1}(9)$ .

Show worked answer

Recognise it: find the input that produces output $9$ .

$f^{-1}(x)=\frac{x+7}{4}$

$f^{-1}(9)=\frac{9+7}{4}=\boxed{4}$

Alternatively, solve the shorter equation $f(x)=9$ :

$4x-7=9\quad\Rightarrow\quad4x=16\quad\Rightarrow\quad\boxed{x=4}$

Find the inverse of a fractional function

Higher only5 marks

Given $f(x)=\frac{x+1}{x-2}$ , find $f^{-1}(x)$ .

Show worked answer

Recognise it: the input appears in both the numerator and denominator, so swap $x$ and $y$ , clear the fraction and collect the $y$ terms.

$y=\frac{x+1}{x-2}$

Swap $x$ and $y$ .

$x=\frac{y+1}{y-2}$

$x(y-2)=y+1$

$xy-2x=y+1$

$xy-y=2x+1$

$y(x-1)=2x+1$

$\boxed{f^{-1}(x)=\frac{2x+1}{x-1}},\qquad x\ne1$

Reverse a conversion function

3 marks

Temperature in degrees Fahrenheit is given by $T(c)=1.8c+32$ , where $c$ is the temperature in degrees Celsius. Find $c$ when $T(c)=68$ .

Show worked answer

Recognise it: the output $68$ is known and the original input is required.

$1.8c+32=68$

$1.8c=36$

$\boxed{c=20^\circ\text{C}}$

Find both inputs to a composite

Higher only5 marks

Given $f(x)=2x+3$ and $g(x)=x^2$ , solve $gf(x)=49$ .

Show worked answer

Recognise it: $gf(x)$ squares the output of f, so a positive and a negative intermediate value are possible.

$gf(x)=(2x+3)^2$

$(2x+3)^2=49$

$2x+3=7\quad\text{or}\quad2x+3=-7$

$\boxed{x=2\quad\text{or}\quad x=-5}$

Examiner-style feedback

Common functions mistakes

Reading f(x) as multiplication

f is the name of the rule. f(3) means apply function f to input 3; it does not mean f multiplied by 3.

Dropping brackets around negatives

For f(x) = x² − 1, write f(−3) = (−3)² − 1. Without brackets, the calculator can apply the square incorrectly.

Applying composite functions left to right

In fg(x) = f(g(x)), g acts first because it is inside. Rewrite the notation as nested brackets.

Substituting only part of a rule

When the input is x + 4, replace every x in the outer rule by the whole expression (x + 4).

Undoing an inverse in the same order

An inverse reverses operations in reverse order. If f doubles then adds 3, f⁻¹ subtracts 3 then halves.

Losing a second quadratic solution

If a squared expression equals 25, the expression can be 5 or −5. Check both resulting inputs.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

f(a): substitute a into every x in the rule.
fg(x): rewrite as f(g(x)) and work from the inside out.
ff(x): apply f to its own output; it means f(f(x)).
f⁻¹(x): reverse the operations in reverse order, or swap input and output and rearrange.
Check notation, brackets and every possible solution before finishing.

Quick answers

Functions FAQ

What does f(x) mean?

It means the output of function f for input x. The letter f names the rule; it is not a variable being multiplied by x.

Which function is applied first in fg(x)?

g is applied first: fg(x) means f(g(x)). Work from the function nearest the input outwards.

How do I find an inverse function?

Reverse the function-machine operations in reverse order. You can confirm the algebra by writing y = f(x), swapping x and y, then rearranging for y.

Are inverse and composite functions Higher tier?

Yes. OCR J560 places inverse and composite processes in its Higher-tier column but explicitly states that knowledge of function notation is not required. AQA and Pearson Edexcel assess formal function notation.

How can I check an inverse?

Choose an input, apply f, then apply f⁻¹ to that output. You should return to the starting input.

Build connected skills

What to revise next

Solve linked rules

Simultaneous equations

Find values that satisfy two algebraic relationships at the same time.

Revise simultaneous equations

Handle powers

Indices

Use power notation confidently inside quadratic and composite functions.

Revise indices

Build a study loop

How to revise

Turn topic gaps into short, measurable GCSE Maths revision sessions.

Revise how to revise

Content standards

Curriculum and rights review

Curriculum references checked 4 September 2026. Simple input-output processes are shared GCSE Mathematics content. Inverse and composite functions are Higher-tier content. OCR J560 6.05a places those processes in its Higher-tier column and explicitly states that knowledge of function notation will not be required; AQA and Pearson Edexcel assess formal notation. All questions, values, contexts and solution wording are original Pass an Exam content.

Official specification references