Domain and Range from a Graph: A Complete Guide
Domain and range are straightforward to describe and easy to get slightly wrong. Almost all the lost marks come from two places: whether an endpoint is included, and whether a gap in the curve has been noticed at all.
What they mean, on the graph
Domain is the set of x-values the function is defined for. Imagine flattening the whole curve down onto the x-axis: the shadow it casts is the domain.
Range is the set of y-values it actually produces. Squash the curve sideways onto the y-axis instead, and that shadow is the range.
For that curve the domain is [2, ∞) and the range is
[0, ∞). Both endpoints are included, because the curve genuinely
reaches the point (2, 0).
Interval notation without the guesswork
Three symbols and one rule cover almost everything.
| Symbol | Means | Use when |
|---|---|---|
[ or ] | Endpoint included | The graph reaches that exact value. A filled dot. |
( or ) | Endpoint excluded | The graph approaches but never arrives. An open dot, or an asymptote. |
∪ | Union, "and also" | The domain or range comes in separate pieces. |
The one rule worth memorising: infinity always gets a round
bracket. (-∞, 5] is correct; [-∞, 5] is
not, because no curve ever reaches infinity.
The families you will be asked about
Polynomials: everything is allowed
Lines, parabolas, cubics and every other polynomial accept any real input,
so the domain is always (-∞, ∞). The range depends on the
degree. An odd-degree polynomial covers every output, so its range is also
(-∞, ∞). An even-degree one has a highest or lowest point that
caps it.
y = x² - 4x + 1 has its vertex at (2, -3), so its
range is [-3, ∞).
Square roots: the inside cannot be negative
Set what is under the root to be at least zero and solve. For
y = √(5 - x), you need 5 - x ≥ 0, so
x ≤ 5 and the domain is (-∞, 5].
Rational functions: the bottom cannot be zero
Find where the denominator is zero and remove exactly those points. For
y = 1/(x² - 9), the denominator vanishes at x = ±3,
so the domain is (-∞, -3) ∪ (-3, 3) ∪ (3, ∞).
Logarithms: the inside must be strictly positive
Note strictly. ln(0) is undefined, so the endpoint is
excluded. y = ln(x - 1) has domain (1, ∞) with a
round bracket. Its range, though, is all real numbers, because a log
function grows without bound in both directions.
Exponentials: everything in, positives out
y = 2ˣ accepts any input, so its domain is
(-∞, ∞). Its output is always positive and never reaches zero,
so its range is (0, ∞) with a round bracket at zero.
Sine and cosine: bounded output
Both accept any input and both produce values between -1 and 1 inclusive, so
the range is [-1, 1]. Square brackets, because they genuinely
reach both extremes. Amplitude changes those numbers:
y = 3sin(x) has range [-3, 3].
Endpoints: the mark-losing details
Filled dot or open dot
A filled dot means the point is part of the graph, so use a square bracket.
An open dot means the curve stops just short, so use a round bracket. If a
printed graph shows an open circle at (4, 2), the value 4 is
not in the domain.
An asymptote is never included
A vertical asymptote marks a value the function cannot take. It is always
excluded, always with a round bracket, and it always splits the domain into
separate pieces joined by ∪.
A turning point is included
The vertex of an upward parabola is the lowest output the function ever produces, and it genuinely produces it, so the range uses a square bracket there.
Watch for a hole
y = (x² - 4)/(x - 2) simplifies to y = x + 2
everywhere except x = 2, where the original is undefined. The
graph looks like a straight line with a single point missing. The domain
excludes 2, and the range excludes 4. This is worth checking whenever a
rational function's numerator and denominator share a factor.
A method that works every time
- Graph it, so you can see the shape rather than reason about it blind.
- Scan left to right for the domain. Where does the curve start? Where does it stop? Are there gaps in the middle?
- Scan bottom to top for the range. What is the lowest point the curve reaches, and the highest?
- Check every boundary and decide included or excluded. This is where the marks are.
- Confirm algebraically. Square roots need a non-negative inside, denominators need a non-zero bottom, logs need a positive inside. If the algebra and the picture disagree, trust the algebra and look again at the picture.
Zoom out before you decide. A curve that looks like it stops at the edge of the window usually continues. Press Home to reset the view, then zoom out once or twice before committing to an answer.
Questions
What is the difference between domain and range?
Domain is every input the function accepts, read left to right along the x-axis. Range is every output it produces, read bottom to top along the y-axis. Domain is what you are allowed to put in; range is what can come out.
When do I use a square bracket instead of a round one?
A square bracket includes the endpoint, a round bracket excludes it. Use square when the graph actually reaches that value, round when it only approaches it or stops just short. Infinity always takes a round bracket, because a curve never reaches infinity.
What is the domain of a square root function?
Whatever makes the inside non-negative. For y = sqrt(x - 3) you need x - 3 to be at least 0, so the domain is x greater than or equal to 3, written [3, infinity).
Does a vertical asymptote get included in the domain?
No. The function is undefined there, so that single value is removed. For y = 1/(x - 2) the domain is everything except 2, written (-infinity, 2) union (2, infinity).
Can the range be a single number?
Yes. A horizontal line such as y = 4 has range {4}, because that is the only output it ever produces, while its domain is all real numbers.