Domain and Range

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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.

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

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

From a graph

  1. Find the leftmost and rightmost points. The domain is every x between them.
  2. Find the lowest and highest points. The range is every y between them.
  3. A closed dot means the end is included. Use \le or \ge.
  4. An open dot means the end is not included. Use \lt or \gt.
  5. 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)\}.

The x-values are -3, 0, 1 and 4. The y-values are 5, 2, 5 and -1. Write 5 only 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.

Left to right, it goes from x = -3 to x = 4. Bottom to top, it goes from y = -1 to y = 3. Both dots are closed, so use \le.

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

Example 3: A graph with an arrow

Two graphs on grids from −5 to 5. Left: a ray starts at a closed dot at (−2, 1) and goes up to the right with an arrow, labeled Domain: x ≥ −2, Range: y ≥ 1. Right: the U-shaped curve y = x² − 4 with its lowest point at (0, −4) and arrows on both ends, labeled Domain: all real numbers, Range: y ≥ −4.
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.

The left graph starts at a closed dot at (-2, 1). The arrow shows it goes up and to the right forever.

So x starts at -2 and never stops growing. y starts at 1 and never stops 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.

You can square any number, so every x works. The lowest point is (0, -4). The arms go up forever, but 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.

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.

The money goes up $12 at a time: 12(0) = 0, 12(1) = 12, and so on, up to 12(40) = 480.

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

Common Mistakes

MistakeExample of the errorFix
Swapping domain and rangeListing the y-values as the domainDomain is x (inputs). Range is y (outputs).
Listing a value twiceRange \{2, 5, 5\}Write each value once: \{2, 5\}.
Mixing up open and closed dotsWriting x \lt 4 for a closed dot at 4Closed dot: \le or \ge. Open dot: \lt or \gt.
Ignoring an arrowWriting -2 \le x \le 3 for a ray that keeps going rightAn arrow means no end: x \ge -2.
Giving a U-shape a range of all numbersSaying y = x^2 - 4 has range all real numbersFind 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.

InputWhat makes senseDomain
Number of people, tickets or shirtsWhole numbers, no negatives0, 1, 2, 3, …
Time after something startsZero or more, parts allowedt \ge 0
Gallons of gas in a 15-gallon tankFrom empty to full0 \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.

  1. \{(2, 7), (4, 9), (6, 7)\}
  2. \{(-1, 0), (-1, 3), (5, 3)\}
  3. A table with x: 0, 1, 2, 3 and y: -4, -2, 0, 2
  4. A segment from (-5, 2) to (1, 6), closed dots at both ends
  5. A segment from a closed dot at (0, -3) to an open dot at (6, 0)
  6. A ray that starts at a closed dot at (3, -2) and goes up and to the left forever
  7. y = 2x - 7
  8. y = 4
  9. y = -x^2 + 5, an upside-down U with its highest point at (0, 5)
  10. 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
  1. Domain \{2, 4, 6\}. Range \{7, 9\}.
  2. Domain \{-1, 5\}. Range \{0, 3\}. (This relation is not a function, but it still has a domain and range.)
  3. Domain \{0, 1, 2, 3\}. Range \{-4, -2, 0, 2\}.
  4. Domain -5 \le x \le 1. Range 2 \le y \le 6.
  5. Domain 0 \le x \lt 6. Range -3 \le y \lt 0.
  6. Domain x \le 3. Range y \ge -2.
  7. Domain: all real numbers. Range: all real numbers.
  8. Domain: all real numbers. Range \{4\}.
  9. Domain: all real numbers. Range y \le 5.
  10. Domain 0 \le g \le 15. Range 0 \le d \le 450 miles, since 30(15) = 450.

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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Next lesson: Key Features of Functions

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