Domain and Range
Identify the domain (all possible inputs) and range (all possible outputs) of a function from graphs, tables, and equations. Represent domain and range using set notation, inequality notation, and interval notation.
A lesson of Algebra 1, in Functions & Relations, about 30 minutes.
By the end of this lesson, you will be able to find the domain and range of a function from its equation, table or graph, write both in inequality and interval notation, and say which real situations restrict them.
What Goes In, and What Comes Out
Every function has two sets attached to it.
- The domain is every input the function will accept
- The range is every output it can produce
For most functions the domain is "any number at all", and the interesting cases are the ones where it is not. Two things stop a number being allowed, and only two:
$$\text{you cannot divide by zero} \qquad \text{you cannot square root a negative}$$
That is the whole list at this level. When a question asks for a domain, you are really being asked: is there a division, is there a square root, and if so what does that forbid?
Range is the harder one, because it depends on what the rule can actually produce rather than on what it will accept. $x^2$ takes any input and returns nothing below zero, however hard you try.
Reading Them Off a Set of Pairs
When a function is a short list, both sets are just the two columns.
$${(1, 4), (2, 7), (5, 7), (8, 2)}$$
$$\text{domain} = {1, 2, 5, 8} \qquad \text{range} = {2, 4, 7}$$
Two things to notice. The range lists $7$ once even though it appears twice, because a set holds each value once. And both are written in increasing order by convention, not because the original order meant anything.
Example 1: From a Table
Here is the whole method on one table, worked all the way through.
| $x$ | $-3$ | $0$ | $2$ | $6$ |
|---|---|---|---|---|
| $f(x)$ | $5$ | $1$ | $1$ | $9$ |
- List the domain, in increasing order, separated by commas -3, 0, 2, 6
- List the range, each value once, in increasing order 1, 5, 9
Worked Example 1: Your Turn
$${(-2, 8), (0, 3), (4, 8), (7, -1)}$$
When the Equation Forbids Something
Now the case that matters. A rule can rule numbers out.
$$f(x) = \frac{1}{x - 3}$$
The denominator cannot be zero, so $x - 3 \neq 0$, so $x \neq 3$. Every other number is fine.
$$\text{domain: all real numbers except } 3$$
$$g(x) = \sqrt{x - 5}$$
You cannot square root a negative, so $x - 5 \geq 0$, so $x \geq 5$.
$$\text{domain: } x \geq 5$$
<!-- SLATE_IF_STRONG: math.advanced-math.function-notation --> <!-- END_SLATE_IF_STRONG -->