Domain and Range of a Function: Examples

The domain of a function is the set of all its inputs, the $x$ -values. The range is the set of all its outputs, the $y$ -values. To find them, list the $x$ – and $y$ -values in a set of points or a table. On a graph, see how far it reaches left to right (domain) and bottom to top (range).

Before this lesson: You should know relations and functions and function notation. Domains on graphs use the inequality signs from compound inequalities.

Extra practice: The Algebra 1 Unit 3 Practice Pack has printable problems, answer keys and a cheat sheet for this unit.

What Are Domain and Range?

Think of a function as a vending machine again. The domain is every button you are allowed to press. The range is every snack that can come out.

On a graph, picture a light shining down from above. The graph casts a shadow on the $x$ -axis. That shadow is the domain. Now shine a light from the side. The shadow on the $y$ -axis is the range.

A line segment on a grid from (−3, −1) to (4, 3), with closed dots at both ends. A green bar on the x-axis runs from −3 to 4 and is labeled Domain: −3 ≤ x ≤ 4. An orange bar on the y-axis runs from −1 to 3 and is labeled Range: −1 ≤ y ≤ 3. — The domain is the graph’s shadow on the x-axis. The range is its shadow on the y-axis.

A memory trick: D comes before R in the alphabet, and x comes before y. So domain goes with x, and range goes with y.

How to Find Domain and Range

From ordered pairs or a table

List every $x$ -value. That list is the domain.
List every $y$ -value. That list is the range.
Write each number only once, from smallest to largest, inside braces { }.

From a graph

Find the leftmost and rightmost points. The domain is every $x$ between them.
Find the lowest and highest points. The range is every $y$ between them.
A closed dot means the end is included. Use $\le$ or $\ge$ .
An open dot means the end is not included. Use $\lt$ or $\gt$ .
An arrow means the graph keeps going forever in that direction.

From an equation or a word problem

For a line like $y = 2x + 1$ , you can put in any number. So the domain is all real numbers. In a word problem, the story sets limits. You can’t sell $-3$ shirts or half a ticket.

Domain and Range Examples

Example 1: A set of ordered pairs

Find the domain and range of $\{(-3, 5), (0, 2), (1, 5), (4, -1)\}$ .

Step 1. List the $x$ -values, the first number in each pair: $-3$ , 0, 1 and 4. They form the domain.

Step 2. List the $y$ -values, the second number in each pair: 5, 2, 5 and $-1$ .

Step 3. Write each output only once, from least to greatest. The 5 appears twice, so write it once.

Answer: Domain $\{-3, 0, 1, 4\}$ . Range $\{-1, 2, 5\}$ .

Example 2: A line segment

Find the domain and range of the segment in the graph above. It runs from $(-3, -1)$ to $(4, 3)$ , with closed dots at both ends.

Step 1. Read the graph from left to right. It starts at $x = -3$ and ends at $x = 4$ .

Step 2. Read the graph from bottom to top. It starts at $y = -1$ and ends at $y = 3$ .

Step 3. Both dots are closed, so the endpoints are included. Use $\le$ on both ends (see compound inequalities).

-3 \le x \le 4

Step 4. Write the domain.

-1 \le y \le 3

Step 5. Write the range.

Answer: Domain $-3 \le x \le 4$ . Range $-1 \le y \le 3$ .

Example 3: A graph with an arrow

Find the domain and range of the ray in the left graph.

Arrows mean the graph keeps going. The ray goes right and up forever. The U-shape goes left and right forever but never below −4.

Step 1. Find where the graph starts. It starts at a closed dot at $(-2, 1)$ , so that point is included.

Step 2. Follow the arrow. It shows the graph goes up and to the right forever.

x \ge -2

Step 3. Read the domain. The $x$ -values start at $-2$ and keep growing.

y \ge 1

Step 4. Read the range. The $y$ -values start at 1 and keep growing.

Answer: Domain $x \ge -2$ . Range $y \ge 1$ .

Example 4: A U-shaped graph

Find the domain and range of $y = x^2 - 4$ , the right graph above.

Step 1. Find the domain. You can square any number, so every $x$ works.

y = 0^2 - 4 = 0 - 4 = -4

Step 2. Find the lowest point. A square is never negative, so $x^2$ is smallest when $x = 0$ .

Step 3. Read the range. The lowest point is $(0, -4)$ . The arms go up forever, so the graph never goes below $-4$ .

Answer: Domain: all real numbers. Range $y \ge -4$ .

Example 5: Word problem

A school club has 40 T-shirts to sell at $12 each. The money earned is $M(s) = 12s$ , where $s$ is the number of shirts sold. Find the domain and range.

Step 1. Find the inputs. The club can sell 0, 1, 2 and so on, up to 40 shirts. It can’t sell part of a shirt, so $s$ is a whole number from 0 to 40.

M(0) = 12(0) = 0

Step 2. Find the smallest output. Selling 0 shirts earns nothing.

M(1) = 12(1) = 12

Step 3. Find the next output. Each shirt adds $12.

M(40) = 12(40) = 480

Step 4. Find the largest output. Selling all 40 shirts earns the most.

Answer: Domain: the whole numbers 0, 1, 2, …, 40. Range: 0, 12, 24, …, 480 dollars.

Common Mistakes

Mistake	Example of the error	Fix
Swapping domain and range	Listing the y-values as the domain	Domain is x (inputs). Range is y (outputs).
Listing a value twice	Range $\{2, 5, 5\}$	Write each value once: $\{2, 5\}$ .
Mixing up open and closed dots	Writing $x \lt 4$ for a closed dot at 4	Closed dot: $\le$ or $\ge$ . Open dot: $\lt$ or $\gt$ .
Ignoring an arrow	Writing $-2 \le x \le 3$ for a ray that keeps going right	An arrow means no end: $x \ge -2$ .
Giving a U-shape a range of all numbers	Saying $y = x^2 - 4$ has range all real numbers	Find the lowest point. The range starts there.

Domain and Range in Real Life

In word problems, ask what inputs make sense. Here are some common limits.

Input	What makes sense	Domain
Number of people, tickets or shirts	Whole numbers, no negatives	0, 1, 2, 3, …
Time after something starts	Zero or more, parts allowed	$t \ge 0$
Gallons of gas in a 15-gallon tank	From empty to full	$0 \le g \le 15$

Some functions have a range of just one number. The horizontal line $y = 4$ has domain all real numbers and range $\{4\}$ . See horizontal and vertical lines.

Practice Problems

Find the domain and range of each function.

$\{(2, 7), (4, 9), (6, 7)\}$
$\{(-1, 0), (-1, 3), (5, 3)\}$
A table with $x$ : 0, 1, 2, 3 and $y$ : $-4$ , $-2$ , 0, 2
A segment from $(-5, 2)$ to $(1, 6)$ , closed dots at both ends
A segment from a closed dot at $(0, -3)$ to an open dot at $(6, 0)$
A ray that starts at a closed dot at $(3, -2)$ and goes up and to the left forever
$y = 2x - 7$
$y = 4$
$y = -x^2 + 5$ , an upside-down U with its highest point at $(0, 5)$
A car goes 30 miles per gallon and its tank holds 15 gallons. The distance it can go is $d(g) = 30g$ , where $g$ is gallons of gas (parts of a gallon allowed).

Show answers

List the x-values: 2, 4 and 6
List the y-values: 7, 9 and 7. Write 7 once
Domain $\{2, 4, 6\}$ . Range $\{7, 9\}$ .

List the x-values: $-1$ , $-1$ and 5. Write $-1$ once
List the y-values: 0, 3 and 3. Write 3 once
Domain $\{-1, 5\}$ . Range $\{0, 3\}$ . (This relation is not a function, but it still has a domain and range.)

Read the x-row: 0, 1, 2 and 3
Read the y-row: $-4$ , $-2$ , 0 and 2
Domain $\{0, 1, 2, 3\}$ . Range $\{-4, -2, 0, 2\}$ .

Left to right, x goes from $-5$ to 1
Bottom to top, y goes from 2 to 6
Both dots are closed, so use $\le$ on both ends
Domain $-5 \le x \le 1$ . Range $2 \le y \le 6$ .

Left to right, x goes from 0 to 6
Bottom to top, y goes from $-3$ to 0
The dot at 0 is closed, so use $\le$ . The dot at (6, 0) is open, so use $\lt$
Domain $0 \le x \lt 6$ . Range $-3 \le y \lt 0$ .

The ray starts at the closed dot $(3, -2)$
It goes to the left forever, so x gets smaller without stopping
It goes up forever, so y gets larger without stopping
Domain $x \le 3$ . Range $y \ge -2$ .

This is a slanted line. You can put in any x
A slanted line goes up and down forever, so it reaches every y
Domain: all real numbers. Range: all real numbers.

This is a horizontal line. You can put in any x
Every output is 4
Domain: all real numbers. Range $\{4\}$ .

You can square any number, so every x works

y = -(0)^2 + 5 = 5

Find the highest point.

The arms go down forever, so the graph never goes above 5
Domain: all real numbers. Range $y \le 5$ .

10.

0 \le g \le 15

Find the inputs: the tank holds 0 to 15 gallons, and parts of a gallon are allowed.

d(0) = 30(0) = 0

Find the smallest output.

d(15) = 30(15) = 450

Find the largest output.

Domain $0 \le g \le 15$ . Range $0 \le d \le 450$ miles.

Want more practice? The Algebra 1 Unit 3 Practice Pack has 39 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

Domain and Range FAQ

What is the domain and range of a function?

The domain is every input ( $x$ -value) the function can use. The range is every output ( $y$ -value) it gives.

How do you find domain and range from a graph?

Read the domain from left to right and the range from bottom to top. Check whether each end has a closed dot, an open dot or an arrow.

What is the domain of a linear function?

All real numbers, unless the line is vertical or a word problem limits it. The range is also all real numbers, except for a horizontal line, whose range is one number.

Is the domain x or y?

The domain is the $x$ -values, and the range is the $y$ -values. D comes before R, just like x comes before y.

Summary

Domain: all the inputs ( $x$ -values). Range: all the outputs ( $y$ -values).
For points or tables, list each value once in braces.
On a graph, read the domain left to right and the range bottom to top.
Closed dot: $\le$ or $\ge$ . Open dot: $\lt$ or $\gt$ . Arrow: no end.
In word problems, the story limits the domain.

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