Domain and Range

Sommaire

In short: Domain and range describe a function’s two sets of numbers: the domain is every input x allowed, and the range is every output y that actually results.

Algebra 1Functions and GraphsF-IF.A.1

What is Domain and Range?

Every function takes inputs and returns outputs, and the pair domain and range says which ones are possible. The domain is the set of x-values you may feed in; the range is the set of y-values that come out.

On a graph, read the domain from left to right along the x-axis, and the range from bottom to top along the y-axis. Restrictions come from the situation, such as hours between 0 and 40, or from the math, such as never dividing by zero or taking the square root of a negative number.

How to spot it

A question asks which inputs make sense, or what values the output can take.
A set of ordered pairs or a table is given and the x-values and y-values must be listed.
A graph stops at endpoints or has a lowest or highest point.
An expression has a denominator or a square root, which restricts the inputs.

The method

List or scan all the x-values that are allowed: that set is the domain.
Exclude any x that causes division by zero or a negative number under a square root.
For the range, find the smallest and largest outputs, or list all the y-values in a table.
Write both using inequalities, interval notation or a list, and check endpoints.

f(x) = x² – 4 on the interval from x = -1 to x = 3.

Domain and Range: solved example

Problem

A ski lodge pays $15 per hour and employees work between 0 and 40 hours a week. For the pay function $P(h) = 15h$, find the domain and the range.

Solution

Hours can be any number from 0 to 40, so the domain is $0 \le h \le 40$.

The pay grows with hours: $P(0) = 0$ and $P(40) = 15 \times 40 = 600$. The range is $0 \le P \le 600$ dollars.

Try it

Question 1

List the domain and range of the relation $\{(2, 5), (4, 9), (6, 5), (8, 1)\}$.

Show the answer

Domain: $\{2, 4, 6, 8\}$. Range: $\{1, 5, 9\}$, listing 5 only once.

Question 2

What value of x is excluded from the domain of $f(x) = \frac{1}{x – 7}$?

Show the answer

$x = 7$, because it makes the denominator zero. The domain is all real numbers except 7.

Domain and Range vs. Function

A function is the rule that connects each input to exactly one output. The domain and the range are the two collections of numbers that the rule works with: where it starts and where it lands. Change the allowed inputs and you can change the range even though the rule stays the same.

Function

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Whole dictionary: A–Z · Algebra 1 · Functions and Graphs

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