Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who want domain and range — the valid inputs and outputs of a function — to become a reliable source of marks instead of a topic they only half-remember.
What query it owns: how to understand and revise domain and range of functions in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the domain and range revision-guide angle, while Tutopiya’s Domain and Range subtopic page owns the learning resource and the free domain and range quiz owns the practice.

Domain and range describe which values a function can accept and produce. In Cambridge IGCSE Mathematics (0580/0607) they appear alongside function notation f(x), graphs and inverse functions. Examiners test whether you can state restrictions — such as denominators not zero or square roots of negatives. This guide explains the subtopic, the exam wording, and where to practise.

Key takeaways

• The domain is the set of allowed input values (usually x).
• The range is the set of possible output values (usually f(x) or y).
• Denominator ≠ 0 and expression under √ ≥ 0 are the two main restrictions.
• Read domain and range from a graph by looking at horizontal and vertical extent.

What are domain and range in Cambridge IGCSE Maths?

The domain of a function is the complete set of values that can be substituted into the function rule. The range is the complete set of values the function can output. In Cambridge IGCSE Mathematics you find them from algebraic rules, from graphs, and in context questions where realistic values are restricted (e.g. lengths must be positive).

Study the notes on Tutopiya’s Domain and Range of Functions subtopic page before attempting questions.

The core ideas you must master

Idea	What it means	How the exam uses it
Domain	Allowed x-values	”State the domain of f(x) = 1/(x − 2)“
Range	Possible f(x) values	”Write down the range of f(x) = x² + 3”
Restriction: denominator	x cannot make denominator 0	”x ≠ 2”
Restriction: square root	Expression under √ must be ≥ 0	”x ≥ −3”
From a graph	Horizontal spread = domain; vertical spread = range	”Write down the domain and range from the graph”

How to find the domain — step by step

1. Identify restrictions: denominators ≠ 0; even roots need ≥ 0 inside.
2. Solve inequalities for each restriction.
3. Combine restrictions and write the domain using inequality or set notation as requested.
4. Check context: if x is a length, x > 0 may apply.
5. From a graph: domain is the x-values covered by the curve (left to right).

Test yourself with the free Domain and Range quiz.

How to find the range

1. From a rule: consider the shape — e.g. x² ≥ 0 so x² + 3 ≥ 3.
2. From a graph: read the lowest and highest y-values the curve reaches.
3. For quadratics, the range depends on whether the parabola opens up or down and the turning point.
4. Fractions such as 1/x typically exclude 0 from the range.
5. Write using inequality notation unless the question asks otherwise.

Domain and range in past-paper wording: command words that matter

Command word / phrase	What the question wants	Typical stem
State the domain	List allowed x-values with restrictions	”State the domain of f(x) = √(x + 4).”
Write down the range	Possible output values	”Write down the range of f(x) = 5 − x².”
f is defined for …	Implicit domain restriction	”f is defined for −2 ≤ x ≤ 3. Find …”
Sketch the graph	Shows domain and range visually	”Sketch y = 1/x for x > 0.”
Explain why x cannot equal …	Link to denominator or root	”Explain why x cannot be 4.”

Worked exam-style stems (how to answer the wording)

1. “State the domain of f(x) = 1/(x − 3).” Denominator ≠ 0 → x ≠ 3 (all real x except 3). Reward: correct exclusion.
2. “Write down the range of f(x) = x² + 2.” x² ≥ 0 → f(x) ≥ 2 → range: f(x) ≥ 2. Reward: correct inequality.
3. “f(x) = √(5 − x). State the domain.” 5 − x ≥ 0 → x ≤ 5. Reward: inequality solved correctly.
4. “The graph of y = f(x) passes from x = −2 to x = 4. Write down the domain.” −2 ≤ x ≤ 4 (or −2 ≤ x ≤ 4 depending on whether endpoints are included). Reward: reading horizontal extent from the graph.
5. “Explain why x = 0 must be excluded from f(x) = 3/x.” Division by zero is undefined, so x ≠ 0. Reward: linking exclusion to denominator — a 1-mark reasoning stem before harder function questions.

Work exam-style questions on the Functions topical past papers resource once the wording is familiar.

How domain and range connect to the Functions unit

Domain and range underpin Graphs of Functions and Quadratic Functions. The Cambridge IGCSE Maths resource hub links every Functions subtopic.

Common mistakes students make

• Confusing domain (x) with range (y or f(x)).
• Forgetting x ≠ a when a value makes the denominator zero.
• Writing x > −4 when the question needs x ≥ −4 for a square root.
• Stating range as a single number instead of an inequality or set.
• Reading domain from the y-axis instead of the x-axis on a graph.

When you need more support

If domain and range questions keep slipping, retake the Domain and Range quiz, practise on Graphs of Functions, and ask a Cambridge IGCSE Maths tutor for help.

Frequently asked questions

What is the domain of a function? The set of input values (usually x) for which the function rule is defined — no zero denominators or negative values under even roots.

What is the range of a function? The set of all output values the function can produce — read from the rule or from the vertical extent of the graph.

How do I find the domain of f(x) = √(x − 2)? Require x − 2 ≥ 0, so x ≥ 2.

How do I revise domain and range effectively? Practise algebraic restrictions first, then reading from graphs. Use the domain and range quiz after each stage.

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