Determining Domain and Range Using the Graph

A function's graph is a direct visual record of its domain and range. The domain is the shadow of the graph on the $x$-axis, and the range is the shadow on the $y$-axis. This section develops that idea through precise procedures and worked examples.

Quick Reference

To find	Draw a line	Interpret
Domain	Vertical lines sweeping all $x$	Domain = $x$-values where vertical lines hit the graph
Range	Horizontal lines sweeping all $y$	Range = $y$-values where horizontal lines hit the graph

Finding Domain and Range from a Graph

A diagram showing how to determine the domain (projection to x-axis) and range (projection to y-axis) from a function's graph. — The domain is the projection of the graph onto the $x$-axis; the range is its projection onto the $y$-axis.

To find the domain: for each $x$-value, draw a vertical line. If it intersects the graph, that $x$ is in the domain. The domain is the set of all $x$-values that produce an intersection.
To find the range: for each $y$-value, draw a horizontal line. If it intersects the graph, that $y$ is in the range. The range is the set of all $y$-values that produce an intersection.

The function $g$ is graphed below. Find its domain and range.

A step-like graph of a function g(x) with a hole at x=1. — Graph of $y = g(x)$.

Solution

A vertical line intersects the graph of $g(x)$ for every $x$ except $x = 1$ (there is a hole at $x = 1$). So $x = 1$ is not in the domain. Horizontal lines intersect the graph only at $y = 1$ and $y = 2$. Therefore: $\operatorname{Dom}(g) = \{x \mid x \neq 1\} = \mathbb{R} - \{1\} = (-\infty, 1) \cup (1, \infty),$ $\operatorname{Rng}(g) = \{1, 2\}.$

The function $\phi$ is graphed below. Determine the domain and range of $\phi$.

A graph of a piecewise function φ(x) consisting of a horizontal ray for negative x and a rising curve for non-negative x. — Graph of $y = \phi(x)$.

Solution

A vertical line through any $x$ always hits either the left piece or the right piece of the graph. Therefore, every real number is in the domain: $\operatorname{Dom}(\phi) = \mathbb{R}.$ Horizontal lines through $y \geq 0$ all intersect the right piece, and the horizontal line $y = -1$ intersects the left piece. No horizontal line through $y < 0$ and $y \neq -1$ hits the graph. Therefore: $\operatorname{Rng}(\phi) = [0, \infty) \cup \{-1\}.$

Find the range of $f : [-1, 2] \to \mathbb{R}$ where $f(x) = x^2 - 3$.

Solution

We sketch the graph by building a table of values, making sure to include the endpoints $x = -1$ and $x = 2$:

$x$	$-1$	$-0.5$	$0$	$0.5$	$1$	$1.5$	$2$
$f(x)$	$-2$	$-2.75$	$-3$	$-2.75$	$-2$	$-0.75$	$1$

Graph of y = x² - 3 on the interval [-1, 2]. — Graph of $y = x^2 - 3$ on $[-1, 2]$.

The function reaches its minimum value of $-3$ at $x = 0$ and its maximum value of $1$ at $x = 2$. From the graph: $\operatorname{Rng}(f) = [-3, 1].$

Frequently Asked Questions

How do I find the domain when the graph has a hole?

A hole means the function is not defined at that $x$-value. Exclude it from the domain. For example, a hole at $x = 1$ means $x = 1 \notin \operatorname{Dom}(f)$, and the domain is $\mathbb{R} - \{1\}$.

Can the range be a union of intervals?

Yes. For example, if the graph consists of two separate pieces, the range is the union of the $y$-values covered by each piece. This is common for piecewise functions.

What if the graph extends infinitely?

Project the infinite portion onto the axis. For example, if the graph extends upward without bound, the range includes $(M, \infty)$ for some value $M$. If it extends to the right without bound, the domain includes $(a, \infty)$ for some $a$.

Is the range always an interval?

Not necessarily. The range can be a finite set (for a piecewise constant function), a union of intervals, or a single point. However, for continuous functions defined on an interval, the range is always an interval (by the Intermediate Value Theorem from calculus).

Determining Domain and Range Using the Graph

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Finding Domain and Range from a Graph

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