Domain & Range from Graphs and Equations

The domain of a function is every input it will accept — the allowed x-values. The range is every output it can produce — the y-values you get back. Together they describe where a function lives.

From a graph you read domain left-to-right and range bottom-to-top. From an equation you hunt for the inputs that break it: division by zero and even roots of negatives are the two usual troublemakers.

Reading domain and range off a graph

Scan the graph horizontally for the domain: what is the leftmost x the curve reaches, and the rightmost? Scan vertically for the range: the lowest and highest y-values.

Watch the endpoints. A filled dot includes that value; an open dot or an arrow that keeps going means the set continues. Arrows in both directions usually mean all real numbers.

The parabola below stretches left and right forever (domain = all real numbers) but bottoms out at $y = -2$ , so its range is $y \geq -2$ .

Finding domain from an equation

Most functions accept every real number. The exceptions are fractions, where the denominator cannot be zero, and even roots, where the inside cannot be negative.

For $f(x) = \dfrac{7}{x + 4}$ , the denominator is zero at $x = -4$ , so the domain is all real numbers except $-4$ . Solve 'denominator = 0' or 'inside of root $\geq 0$ ' to find the restriction.

Worked examples

Example 1: domain of a rational function

Find the domain of $f(x) = \dfrac{7}{x + 4}$ .

The denominator cannot be zero

x + 4 \neq 0

Solve for the excluded value

x \neq -4

State the domain

\text{all reals except } -4

Answer: All real numbers except $x = -4$

Example 2: domain of a square root

Find the domain of $g(x) = \sqrt{x - 2}$ .

The inside must be at least zero

x - 2 \geq 0

Solve

x \geq 2

Answer: $x \geq 2$

Example 3: domain and range from a graph

A parabola opens upward with arrows on both ends and its lowest point at $(0, -2)$ . Find its domain and range.

Scan left to right; both arrows keep going

\text{all real numbers}

Scan bottom to top; the lowest y reached is -2

y = -2

The graph rises from there, and -2 is included

y \geq -2

Answer: Domain: all real numbers; range: $y \geq -2$

Try one yourself

Common questions

Which functions have restricted domains?

Mainly fractions (denominator cannot be zero) and even roots like square roots (the inside cannot be negative). Polynomials accept every real number.

How do I read range from a graph with an arrow?

An upward arrow means the y-values continue without bound, so the range goes to infinity. Read the lowest reached y-value as the other end.

Is domain always the x-values?

Yes. Domain is inputs (x), range is outputs (y). A quick way to remember: domain comes first alphabetically and x comes first on the axes.

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Domain & Range from Graphs and Equations

Reading domain and range off a graph

Finding domain from an equation

Worked examples

Example 1: domain of a rational function

Example 2: domain of a square root

Example 3: domain and range from a graph

Try one yourself

Common questions

Keep learning

Interval & Set-Builder Notation

Key Features of Functions

Function Notation, Evaluating & Interpreting