Functions: domain, range and harder inverses

A function has inputs (the domain) and outputs (the range). Some inputs must be left out, and some inverses need the 'subject appears twice' method.

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Domain and range

The domain is the set of inputs; the range is the set of outputs.
f(x) = x2 + 3 for all x: the smallest output is 3, so the range is f(x) ≥ 3.

Values to exclude

You cannot divide by zero: f(x) = 5(x + 4) needs x ≠ −4.
You cannot square-root a negative: g(x) = √(x − 3) needs x ≥ 3.

Harder inverses and composites

f(x) = (2x + 1)(x − 3). Write x = (2y + 1)(y − 3) and make y the subject:
x(y − 3) = 2y + 1, so xy − 2y = 3x + 1, y(x − 2) = 3x + 1
f−1(x) = (3x + 1)(x − 2)
For a composite, substitute the whole of the inner function: if f(x) = 2x + 1 and g(x) = x2, fg(x) = 2x2 + 1.
Worked example

f(x) = 3x − 1. Find ff(2).

  1. f(2) = 3 × 2 − 1 = 5
  2. f(5) = 3 × 5 − 1 = 14

Answer: ff(2) = 14

Key idea

Exclude inputs that make you divide by zero or square-root a negative. For an inverse, swap x and y and make y the subject, factorising if y appears twice.

Key words
Range
The largest value minus the smallest value. It measures spread.
Function
A rule that turns each input into one output, written f(x).

All GCSE and International GCSE Mathematics key terms and definitions →

Extra practice

2 more questions

Try each one, then tap to check.

  1. Which value of x must be excluded from the domain of f(x) = 7(x − 7)?

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    Answer: 7

    x − 7 = 0 when x = 7, and you cannot divide by zero.

  2. f(x) = x2 + 10 for all values of x. What is the smallest value in the range of f?

    Show the answer

    Answer: 10

    x2 is never negative, so the smallest output is 0 + 10 = 10.

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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