What is the domain of a function in Mathematics?

In Mathematics, particularly in the Cambridge IGCSE curriculum, the domain of a function refers to the complete set of possible input values (x-values) for which the function is defined. For example, the domain of the function f(x) = 1/x is all real numbers except x = 0, because division by zero is undefined.

How do you determine the range of a function?

To determine the range of a function, you need to identify all possible output values (y-values) that the function can produce. This often involves analyzing the function's equation and considering any restrictions on the output. For instance, the range of f(x) = x^2 is all non-negative real numbers, as squaring any real number cannot result in a negative value.

Why is understanding the domain and range important in the Cambridge IGCSE Mathematics curriculum?

Understanding the domain and range is crucial in the Cambridge IGCSE Mathematics curriculum because it helps students comprehend the behavior and limitations of functions. This knowledge is essential for solving problems involving functions, graphing them accurately, and applying them in real-world contexts.

Can a function have an infinite domain or range?

Yes, a function can have an infinite domain or range. For example, the function f(x) = x has an infinite domain and range, as it is defined for all real numbers and can produce any real number as an output. Understanding these concepts is key in the study of functions within the Cambridge IGCSE Mathematics syllabus.

What is the difference between the domain and range of a function?

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values) that the function can produce. In the context of Cambridge IGCSE Mathematics, distinguishing between these two concepts is fundamental for analyzing and graphing functions accurately.

Why is "Including values that make the denominator zero" a common mistake in Domain and Range of Functions?

Why it happens: Students forget to exclude x = a when g(a) = 0. How to avoid it: Set the denominator equal to zero, solve, and exclude that x. Source: 0580/42 — recurring

Why is "Writing $g(x) > 0$ instead of $g(x) \geq 0$ for a square root" a common mistake in Domain and Range of Functions?

Why it happens: Confusing strict and non-strict inequalities. How to avoid it: √(0) = 0 is defined, so use ≥ 0.

Why is "Confusing domain with range" a common mistake in Domain and Range of Functions?

Why it happens: Both involve sets of numbers; students mix which is which. How to avoid it: Domain = INPUTS (x). Range = OUTPUTS (f(x) or y).

Why is "Stating the range of a quadratic without finding the vertex" a common mistake in Domain and Range of Functions?

Why it happens: Students compute f at the endpoints of the domain, missing that the minimum (or maximum) lies between them. How to avoid it: Complete the square or use -b/2a to find the vertex; that's where the extreme value lives.

Cambridge IGCSE Mathematics (0580)

Domain and Range of Functions

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Short Notes - Domain and Range of Functions

Detailed Study Notes

Detailed notes on Functions for Cambridge IGCSE Mathematics, covering key concepts, explanations, examples, and exam-focused revision points.

Domain and Range — Cambridge IGCSE 0580 Maths Extended (2026)

The domain is what you put IN; the range is what you get OUT. The two sets that define a function — and a recurring source of marks on Paper 4 once you can think graphically.

At a glance

Domain: the set of allowed inputs.
Range: the set of all possible outputs.
$f(x) = \dfrac{1}{x}$ : domain excludes $0$ .
$f(x) = \sqrt{x}$ : domain $x \ge 0$ , range $f \ge 0$ .
$f(x) = x^{2}$ : domain all reals, range $f \ge 0$ .
Read range from a graph: project all y-values the curve hits.
Quadratic range: $y \ge \text{vertex}$ (if opens up) or $y \le \text{vertex}$ (opens down).

What you’ll learn

Mapped to the Cambridge IGCSE 0580 syllabus (2025-2027).

E2.15 — Use function notation; identify domain and range; sketch graphs of common functions.

What's a domain?

The set of inputs the function ACCEPTS. Cambridge usually states it, or expects you to find it from the formula.

The domain of a function $f$ is the set of all input values $x$ for which $f(x)$ is defined.

The domain is everything you're allowed to feed in; the range is everything that can come out.

Common natural domains:

Polynomials ( $f(x) = x^{2}, x^{3}, \ldots$ ): domain is ALL real numbers $\mathbb{R}$ .
Rational functions ( $f(x) = \dfrac{1}{x - 2}$ ): exclude any $x$ that makes the denominator zero. Here $x \ne 2$ .
Square root: $f(x) = \sqrt{x - 3}$ : need $x - 3 \ge 0$ , so $x \ge 3$ .
Logarithms: $f(x) = \log x$ : $x > 0$ .

Only inputs from 3 upward keep x − 3 non-negative, so the domain is x ≥ 3.

Cambridge often gives a RESTRICTED domain like $f(x) = x^{2}$ for $0 \le x \le 3$ — i.e. only that piece of the parabola.

Notation. Domain stated as an inequality or interval: $x \in [-2, 5]$ means $-2 \le x \le 5$ .

Domain = allowed inputs.
Polynomial: all reals.
Rational: exclude zeros of the denominator.
Square root: argument $\ge 0$ .

What's a range?

The set of OUTPUTS the function actually produces. Read it off a sketch.

The range of a function is the set of all possible output values: $\{f(x) : x \in \text{domain}\}$ .

Common ranges.

Function	Range
$f(x) = x^{2}$	$f(x) \ge 0$
$f(x) = \sqrt{x}$	$f(x) \ge 0$
$f(x) = \dfrac{1}{x}$	$f(x) \ne 0$
$f(x) = \sin x$	$-1 \le f(x) \le 1$
$f(x) = e^{x}$	$f(x) > 0$
$f(x) = \log x$	all reals

Method to find the range.

Sketch the graph (or visualise mentally).
Identify the lowest and highest $y$ -values the curve actually hits over the given domain.
State as an inequality: $a \le f(x) \le b$ .

Worked. Find the range of $f(x) = x^{2} + 1$ for $-2 \le x \le 3$ .

The minimum is at the vertex inside the domain; the maximum is at the further endpoint.

Curve is a parabola with vertex at $(0, 1)$ — minimum.
Endpoint values: $f(-2) = 5$ , $f(3) = 10$ .
Range: $1 \le f(x) \le 10$ (minimum is at the vertex inside the domain; max is at the further endpoint).

Worked. Range of $f(x) = -x^{2} + 4$ over all reals.

Parabola opens DOWN. Vertex at $(0, 4)$ — maximum.
$f(x) \le 4$ .

Range = outputs.
Sketch first; read off y-values.
Watch for vertex inside the domain (max/min).
Always include endpoints if they're hit.

Reading domain and range from a graph

Project the graph onto the axes — the shadow on $x$ -axis is the domain; on $y$ -axis, the range.

Imagine projecting the graph onto each axis with vertical lines (for domain) and horizontal lines (for range).

Shadow onto the x-axis = domain; shadow onto the y-axis = range.

Domain. All $x$ -values where the curve exists. From a graph defined on $-3 \le x \le 5$ , the domain is $[-3, 5]$ .

Range. All $y$ -values the curve hits. From the same graph, look at the lowest and highest $y$ the curve reaches and any gaps.

Discontinuous functions. If a function has a break (e.g. $\dfrac{1}{x}$ near zero), the domain has a GAP and the range may have a GAP too.

Piecewise functions. Each piece contributes its own segment to the range; combine them.

Project onto $x$ -axis: domain.
Project onto $y$ -axis: range.
Discontinuities create gaps in domain or range.
Always check endpoints whether they're included (closed circle) or excluded (open circle).

Quick recap

Domain = inputs. Range = outputs.
Polynomials: all reals. Square root: $\ge 0$ . Reciprocal: $\ne 0$ .
Sketch the graph, then read off the range.
Watch out for vertex (max/min) inside a restricted domain.
Project onto each axis: $x$ -shadow → domain, $y$ -shadow → range.

Memorise this

Verbatim phrases and definitions Cambridge mark schemes credit.

Domain — the set of all valid input values for a function.
Range — the set of all output values the function actually takes.
Function — a rule that assigns exactly one output to each input.

How it’s examined

Domain and range appear most years on Paper 4 as a 2-3 mark item, usually paired with sketching a graph or reading from one. Examiner reports flag forgetting to check the vertex of a parabola when computing the range over a restricted domain.

Sources: Cambridge IGCSE Mathematics 0580 syllabus 2025-2027 (E2.15); 0580/42 Oct/Nov 2024 — Q12 (range of restricted quadratic); 0580 Examiner Reports 2022-2024. Last reviewed 2026-05-05.

Take this whole topic with you

Step-by-step worked examples — Domain and Range of Functions

Step-by-step solutions to past-paper-style questions on domain and range of functions, written exactly the way a tutor would explain them at the board.

Question type:

Question patterns to master — Domain and Range of Functions

Almost every domain and range of functions exam question is one of these shapes. Learn to spot each one and you will always know how to start.

Direct calculation

Recognise it by

A single instruction — state the domain, find the range, find $f(-2)$ — applied to one given function with one method.

How to approach it

Match the function type to its rule: rational ⇒ denominator $\neq 0$ ; square root ⇒ radicand $\geq 0$ ; quadratic ⇒ complete the square to find the vertex; linear on a closed interval ⇒ test the endpoints.

Common trap

Stating the range of a quadratic straight from the endpoints without finding the vertex, or writing $x > 0$ instead of $x \geq 0$ for a square root. Examiner reports flag both as recurring slips.

Graph or diagram

Recognise it by

The question gives — or asks you to read — a graph or sketch, quoting features such as a stated minimum, maximum or turning point rather than a formula.

How to approach it

Read the domain off the $x$ -axis and the range off the $y$ -axis. On a closed continuous interval the function takes every value from the lowest $y$ (minimum) to the highest $y$ (maximum).

Common trap

Reading the $x$ -coordinates of the turning points instead of the $y$ -coordinates — that gives the domain, not the range. Quote $y$ -values for the range.

Multi-step problem

Recognise it by

Several stages chained together — restrict a domain then invert, or form a composite then find its range.

How to approach it

Do one stage at a time: complete the square or form the composite first, then apply the domain/range or inverse rule to that result, carrying the restriction through.

Common trap

Forgetting to evaluate the inner function at its minimum before reading the composite's range, or losing the domain restriction when choosing the root of an inverted quadratic.

1Find the domain of a rational function
ExtendedDirect calculation• domain, rational
Question
State the domain of $f(x) = \dfrac{1}{x - 3}$ .
Step-by-step solution
1. Step 1
  Domain is all real $x$ EXCEPT values that make the denominator zero.
  $x - 3 \neq 0 \implies x \neq 3$
Answer
Domain: $x \in \mathbb{R},\ x \neq 3$
2Domain of a square-root function
ExtendedDirect calculation• Adapted from 0580/42 May/Jun 2024 Q10• domain, square root
Question
State the domain of $g(x) = \sqrt{2x - 5}$ .
Step-by-step solution
1. Step 1
  We need the radicand non-negative.
  $2x - 5 \geq 0 \implies x \geq 2.5$
Answer
Domain: $x \geq 2.5$
3Find the range of a quadratic
ExtendedDirect calculation• range, quadratic
Question
Find the range of $f(x) = x^{2} - 4x + 7$ .
Step-by-step solution
1. Step 1
  Complete the square to find the minimum.
  $f(x) = (x - 2)^{2} + 3$
2. Step 2
  $(x-2)^{2} \geq 0$ , so $f(x) \geq 3$ . The minimum value of $3$ is reached at $x = 2$ .
Answer
Range: $f(x) \geq 3$
4Range from a restricted domain
ExtendedDirect calculation• range, linear
Question
$f(x) = 2x - 3$ for $-1 \leq x \leq 4$ . Find the range.
Step-by-step solution
1. Step 1
  $f$ is linear and increasing, so the range comes from the endpoints.
  $f(-1) = -5,\quad f(4) = 5$
Answer
Range: $-5 \leq f(x) \leq 5$
5Evaluate a function at a value
CoreDirect calculation• function notation
Question
Given $f(x) = 3x^{2} - x + 2$ , find $f(-2)$ .
Step-by-step solution
1. Step 1
  Substitute $x = -2$ with brackets.
  $f(-2) = 3(-2)^{2} - (-2) + 2$
2. Step 2
  Evaluate.
  $= 12 + 2 + 2 = 16$
Answer
$f(-2) = 16$
6Domain of $f(x) = \sqrt{x}$
CoreDirect calculation• domain, square root
Question
State the domain and range of $f(x) = \sqrt{x}$ .
Step-by-step solution
1. Step 1
  For a real value, the radicand must be $\geq 0$ .
  $x \geq 0$
2. Step 2
  The principal square root is non-negative for any non-negative input, and reaches every non-negative value.
  $f(x) \geq 0$
Answer
Domain: $x \geq 0$ ; Range: $f(x) \geq 0$
Examiner tip
The examiner report flags candidates often write $x > 0$ instead of $x \geq 0$ . $\sqrt{0} = 0$ is defined, so use the closed inequality.
7Read the range from a graph
ExtendedGraph or diagram• Adapted from 0580/42 Oct/Nov 2023 Q12• range, graph reading
Question
The graph of $y = f(x)$ has a minimum at $(2, -3)$ and a maximum at $(5, 4)$ , on the domain $0 \leq x \leq 6$ . State the range of $f$ .
Step-by-step solution
1. Step 1
  The range is the set of $y$ -values the function takes. The minimum is the lowest $y$ -value and the maximum is the highest.
2. Step 2
  Because the function is continuous on a closed interval, it takes every value between the min and max.
  $-3 \leq f(x) \leq 4$
Answer
$-3 \leq f(x) \leq 4$
8Range of an absolute-value function
ExtendedDirect calculation• range, absolute value
Question
Find the range of $f(x) = |x - 2| + 1$ on the domain $-3 \leq x \leq 5$ .
Step-by-step solution
1. Step 1
  $|x - 2| \geq 0$ for all $x$ , with equality at $x = 2$ (inside the domain). So the minimum of $f$ is $0 + 1 = 1$ .
2. Step 2
  Evaluate at the endpoints.
  $f(-3) = 5 + 1 = 6,\ f(5) = 3 + 1 = 4$
3. Step 3
  Maximum is $6$ , minimum is $1$ .
Answer
Range: $1 \leq f(x) \leq 6$
9Restrict the domain so the inverse exists
ChallengeMulti-step problem• domain, inverse, one-to-one
Question
$f(x) = x^{2} - 6x + 5$ . State the largest possible domain $x \geq a$ for which $f^{-1}$ exists, and find $f^{-1}(x)$ on that domain.
Step-by-step solution
1. Step 1
  Complete the square to locate the vertex.
  $f(x) = (x - 3)^{2} - 4$
2. Step 2
  $f$ is one-to-one on $x \geq 3$ (the right-hand branch of the parabola). So $a = 3$ .
3. Step 3
  Find $f^{-1}$ by swapping and solving.
  $x = (y - 3)^{2} - 4 \implies y = 3 + \sqrt{x + 4}$
4. Step 4
  Take the positive root because the original domain forced $y - 3 \geq 0$ .
Answer
Largest domain: $x \geq 3$ . Inverse: $f^{-1}(x) = 3 + \sqrt{x + 4}$ .
Examiner tip
The mark scheme awards method marks for justifying the chosen root by referring to the restricted domain — " $x \geq 3 \Rightarrow y - 3 \geq 0$ " is the key statement.
10Range of a composite function
ChallengeMulti-step problem• range, composite
Question
$f(x) = x^{2} + 1$ and $g(x) = \sqrt{x}$ for $x \geq 0$ . Find the range of $gf(x)$ .
Step-by-step solution
1. Step 1
  Form the composite.
  $gf(x) = g(x^{2} + 1) = \sqrt{x^{2} + 1}$
2. Step 2
  For all real $x$ , $x^{2} \geq 0$ , so $x^{2} + 1 \geq 1$ . Then $\sqrt{x^{2} + 1} \geq 1$ .
3. Step 3
  The minimum value $1$ is reached at $x = 0$ , and the expression grows without bound as $|x|$ increases.
Answer
Range: $gf(x) \geq 1$
Examiner tip
The examiner report flags candidates often write " $gf(x) \geq 0$ " by reflex. Always evaluate at the inner minimum first — here $x = 0$ gives the composite minimum of $1$ .

Key Formulae — Domain and Range of Functions

The formulae you need to memorise for domain and range of functions on the Cambridge IGCSE 0580 paper, with every variable defined in plain English and a note on when to use it.

Excluded values (rational)
$f(x) = \dfrac{1}{g(x)} \Rightarrow g(x) \neq 0$
When to use
Whenever the function has a variable in the denominator.
Domain of a square root
$f(x) = \sqrt{g(x)} \Rightarrow g(x) \geq 0$
When to use
Whenever the function has a variable inside an even-index radical.

Key Definitions and Keywords — Domain and Range of Functions

Definitions to memorise and the exact keywords mark schemes credit for domain and range of functions answers — sharpened from recent examiner reports for the 2026 0580 sitting.

Function
Examiner keyword
A relation that assigns each input value exactly one output value.
Domain
Examiner keyword
The set of all input values ( $x$ ) for which the function is defined.
Range
Examiner keyword
The set of all output values ( $f(x)$ ) the function takes for inputs in its domain.
Function notation
$f(x)$ denotes the value of function $f$ at input $x$ . $f: x \mapsto 3x + 2$ is the same as $f(x) = 3x + 2$ .

Common Mistakes and Misconceptions — Domain and Range of Functions

The traps other students keep falling into on domain and range of functions questions — taken from recent Cambridge IGCSE 0580 examiner reports and mark schemes — and how to avoid them.

✕Including values that make the denominator zero
0580/42 — recurring
Why it happens
Students forget to exclude $x = a$ when $g(a) = 0$ .
How to avoid it
Set the denominator equal to zero, solve, and exclude that $x$ .
✕Writing $g(x) > 0$ instead of $g(x) \geq 0$ for a square root
Why it happens
Confusing strict and non-strict inequalities.
How to avoid it
$\sqrt{0} = 0$ is defined, so use $\geq 0$ .
✕Confusing domain with range
Why it happens
Both involve sets of numbers; students mix which is which.
How to avoid it
Domain = INPUTS ( $x$ ). Range = OUTPUTS ( $f(x)$ or $y$ ).
✕Stating the range of a quadratic without finding the vertex
Why it happens
Students compute $f$ at the endpoints of the domain, missing that the minimum (or maximum) lies between them.
How to avoid it
Complete the square or use $-\dfrac{b}{2a}$ to find the vertex; that's where the extreme value lives.