Domain and Range of Continuous Functions

Q: How do I remember which one is domain and which is range?

Domain goes with $x$ and range goes with $y$ — the pairing is alphabetical: $d$ before $r$, $x$ before $y$. Domain is read left to right along the horizontal axis; range is read bottom to top along the vertical axis.

Q: When do I use a bracket and when do I use a parenthesis?

A square bracket means the endpoint is included, matching a filled-in dot and a $\leq$ sign. A parenthesis means the endpoint is not included, matching a hollow dot and a $<$ sign. Infinity always takes a parenthesis.

Q: What if the graph has an arrow on only one end?

Then only that direction goes on forever. A graph starting at a closed point at $x = 0$ with an arrow to the right has domain $x \geq 0$, or $[0, \infty)$ — closed bracket at the start, parenthesis at infinity.

A continuous function is one you can draw without lifting your pencil — a solid line or curve instead of separate dots. For a graph like that, the domain is every $x$ -value the graph covers, and the range is every $y$ -value it reaches. Because the graph is unbroken, both come out as whole stretches of numbers, not lists.

Reading them is a scanning job. For the domain, scan the graph left to right and note where it starts and stops horizontally. For the range, scan bottom to top and note the lowest and highest heights. Everything else in this topic is just how to write those stretches down.

Domain: scan left to right

Imagine flattening the whole graph down onto the $x$ -axis. The shadow it casts is the domain. A segment that starts above $x = -3$ and ends above $x = 4$ has domain $-3 \leq x \leq 4$ — every $x$ -value in between is used, because the graph is unbroken.

For the range, flatten the graph sideways onto the $y$ -axis instead. If the lowest point of the graph sits at height $-1$ and the highest at height $4$ , the range is $-1 \leq y \leq 4$ .

Open and closed endpoints

A closed (filled-in) endpoint means that value is included. An open (hollow) endpoint means the graph gets arbitrarily close to that value but does not include it. In inequalities, included endpoints get $\leq$ and excluded ones get $<$ . The segment above runs from a closed endpoint at $x = -3$ to an open one at $x = 4$ , so its domain is $-3 \leq x < 4$ .

Interval notation says the same thing with brackets: a square bracket includes the endpoint and a parenthesis leaves it out. That domain is $[-3, 4)$ .

Arrows mean forever

An arrow on the end of a graph means it continues in that direction without stopping. If a graph has arrows carrying it left and right forever, the domain is all real numbers, written $(-\infty, \infty)$ . Infinity always gets a parenthesis, never a bracket — the graph never arrives at infinity, so there is no endpoint to include.

The line below runs off both ends with arrows, so it covers every $x$ -value and climbs through every $y$ -value: domain and range are both all real numbers, $(-\infty, \infty)$ .

Worked examples

Example 1: a segment with two closed endpoints

A segment runs from $(-2, 1)$ to $(3, 6)$ , both endpoints filled in. Find the domain and range.

Scan left to right: the graph starts at

x = -2

and ends at

x = 3

-2 \leq x \leq 3

Scan bottom to top: the lowest point is

y = 1

, the highest is

y = 6

1 \leq y \leq 6

Both endpoints are filled in, so both use square brackets

[-2, 3] \text{ and } [1, 6]

Answer: Domain $[-2, 3]$ , range $[1, 6]$

Example 2: one open endpoint

A segment runs from a closed endpoint at $(-1, -2)$ up to an open endpoint at $(2, 4)$ . Find the domain and range.

Left to right: starts at

x = -1

(included), stops just before

x = 2

(excluded)

-1 \leq x < 2

Bottom to top: starts at

y = -2

(included), stops just before

y = 4

(excluded)

-2 \leq y < 4

Bracket on the included side, parenthesis on the excluded side

[-1, 2) \text{ and } [-2, 4)

Answer: Domain $[-1, 2)$ , range $[-2, 4)$

Example 3: arrows on both ends

A line passes through $(0, 2)$ with arrows on both ends. Find the domain and range.

The arrows carry the line left and right forever, so every

x

-value is covered

A slanted line with arrows also climbs and drops forever, so every

y

-value is reached

Write both in interval notation

(-\infty, \infty)

Answer: Domain: all real numbers. Range: all real numbers.

Try one yourself

Common questions

How do I remember which one is domain and which is range?

Domain goes with $x$ and range goes with $y$ — the pairing is alphabetical: $d$ before $r$ , $x$ before $y$ . Domain is read left to right along the horizontal axis; range is read bottom to top along the vertical axis.

When do I use a bracket and when do I use a parenthesis?

A square bracket means the endpoint is included, matching a filled-in dot and a $\leq$ sign. A parenthesis means the endpoint is not included, matching a hollow dot and a $<$ sign. Infinity always takes a parenthesis.

What if the graph has an arrow on only one end?

Then only that direction goes on forever. A graph starting at a closed point at $x = 0$ with an arrow to the right has domain $x \geq 0$ , or $[0, \infty)$ — closed bracket at the start, parenthesis at infinity.

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Domain and Range of Continuous Functions

Domain: scan left to right

Open and closed endpoints

Arrows mean forever

Worked examples

Example 1: a segment with two closed endpoints

Example 2: one open endpoint

Example 3: arrows on both ends

Try one yourself

Common questions

Keep learning

Domain and Range of Linear & Quadratic Functions

Intro to Domain and Range

Graphing Functions

Function Notation