Domain and Range of a Function: How to Find Them

What domain and range mean, the two restrictions that cause almost every limited domain (dividing by zero and square roots of negatives), and how to read both from a graph.

3 min read Supports 4 lessons

Every function has two lists attached. The domain lists every input you are allowed to use. The range lists every output that actually comes out.

For $f(x) = 2x + 1$ , any number can go in and any number can come out, so both are all real numbers. Many functions are pickier.

The two usual restrictions

Usual restrictions

\frac{1}{x - a}:\; x \neq a \qquad\qquad \sqrt{x - a}:\; x \ge a

You cannot divide by zero, and you cannot take the square root of a negative number.

Worked example

Find the domain of $f(x) = \dfrac{3}{x + 4}$ .

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The denominator cannot be zero: $x + 4 \neq 0$ .

Worked example

Find the domain of $g(x) = \sqrt{x - 1}$ .

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The inside of the root must not be negative: $x - 1 \ge 0$ .

Reading them from a graph

On a graph, the domain is how far the curve stretches left to right. The range is how far it stretches up and down. Imagine squashing the graph flat onto each axis: the shadow on the $x$ -axis is the domain, and the shadow on the $y$ -axis is the range.

Domain $x \ge 1$ ; range $y \ge 0$ .

Finding the range

The range is usually harder. Think about what outputs are possible. A square is never negative, so $x^2$ has range $y \ge 0$ . Adding $3$ shifts every output up, so $x^2 + 3$ has range $y \ge 3$ .

$f(x) = (x - 2)^2 - 5$
Find the range.

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Domains from real life

Context can limit a domain too. If $A(r) = \pi r^2$ is the area of a circle, negative radii make no sense, so the domain is $r \ge 0$ even though the formula would accept anything.

Tip

Write domains and ranges in interval notation: "all reals except $-4$ " is $(-\infty, -4) \cup (-4, \infty)$ .

Watch out

Do not confuse the two. The domain is about $x$ (inputs); the range is about $y$ (outputs).

Common questions

How do you find the domain of a function?: Start with all real numbers, then remove any input that would divide by zero or take an even root of a negative number. Also consider any limits the real situation imposes.
What is the range of a function?: It is the set of all outputs the function actually produces. For $f(x) = x^2$ the range is $y \ge 0$ , since a square is never negative.

Practise domain and range

The Domain & Range lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Lessons on this

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Ideas this article builds on or leads to.

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Common questions

How do you find the domain of a function?

Start with all real numbers, then remove any input that would divide by zero or take an even root of a negative number. Also consider any limits the real situation imposes.

What is the range of a function?

It is the set of all outputs the function actually produces. For $f(x) = x^2$ the range is $y \ge 0$ , since a square is never negative.