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Domain and Range Calculator

DOMAIN AND RANGE
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Type the three coefficients, then press =.
Every condition, and where the range comes from
Both sets in interval notation — exact, or refused by name.
What goes in, and what comes out
Type the function of x on one line. Use ^ for powers, / for division, and brackets wherever you would write them: sqrt(x - 4), 1/(x^2 - 9), (2x+1)/(x-3). The domain is worked out for anything this page can read. The range is given only where an exact answer exists — and named plainly where it does not, rather than guessed at.
f(x) =
Write it as you would on paper — sqrt(x-4), 1/(x^2-9).
Fractions and decimals both work, and nothing is rounded.
Tap a box, then use the keys
The domain is every x the function will accept; the range is every y it can produce. The domain is the easy half — three rules cover almost all of it. The range is the hard half, and most tools quietly change the subject. This one gives it exactly where an exact answer exists and says so plainly where it does not. It also does the two things the textbooks mention and the calculators skip: a hole can take a number out of the range as well, and a curve is sometimes allowed to sit on its own horizontal asymptote.
The working, step by step
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This domain and range calculator gives both sets in interval notation, solved exactly — and tells you plainly when no exact answer exists rather than guessing at one.

What Domain and Range Mean

The domain is every x the function will accept. The range is every y it can produce. One is a checklist; the other is a real question.

Three Rules Cover Almost Everything

Scan the expression for exactly three things:

bottom ≠ 0  inside an even root ≥ 0  inside a log > 0

Solve each and keep the x that satisfy all of them. √(x − 4) needs x ≥ 4, so the domain is [4, ∞). A function with none of the three takes every real number.

The brackets in that answer are doing work. A square bracket means the endpoint is included; a round one means it is not, and infinity always gets a round bracket because you never arrive there. So √(x − 4) gives [4, ∞) — 4 itself is fine, since √0 is 0 — while 1/(x − 4) gives (−∞, 4) ∪ (4, ∞), where 4 is the one number barred. The ∪ joins the pieces that survive. Getting a bracket the wrong way round is a mark lost for something that is not really algebra at all.

The Harder Half

There is no checklist for the range, and textbooks admit as much: for most fractions you cannot pin it down without calculus. Be suspicious of any tool that answers instantly for everything. This one refuses by name instead.

One method does work. Write y = f(x), clear the fraction, and treat it as an equation in x: wherever that has a real solution, y is in the range. For a quadratic in x that means its discriminant is not negative, and the inequality in y that falls out is the answer.

A Hole Costs You a Number Twice

When a factor cancels, the gap is at one x — and the y the curve would have had is gone too:

(x² − 4)/(x − 2) → x ≠ 2 and y ≠ 4

Most working stops at the x. But if some other x gives that same y, the range keeps it: in (x³ − x)/(x − 1) the hole sits at y = 2, and x = −2 gives 2 as well, so nothing leaves. That second check is where a careless mark goes.

Sitting on Your Own Asymptote

"The range excludes the horizontal asymptote" is remembered as a rule and is only half true. It holds for a fraction of two straight lines. It fails the moment the bottom outranks the top: x/(x² + 1) runs to zero at both ends and passes through zero in the middle, so its range is [−½, ½] with zero firmly inside. An asymptote describes what happens far away, not what is forbidden.

Using the Domain and Range Calculator

Type the function on one line, brackets as you would write them. Every condition is solved exactly, so an irrational boundary is refused rather than rounded. For fractions alone, the domain of a rational function page goes deeper.

Domain and Range FAQ

How do you find the domain of a function?
Look for three things only: a denominator, which cannot be zero; an even root, whose inside cannot be negative; and a log, whose inside must be above zero. Solve each one and keep the x that satisfy all of them.
Why is the range harder to find than the domain?
The domain is a checklist. The range asks what the whole function can reach, and for most shapes there is no rule to follow. The reliable method is to write y equals f of x and solve for x: wherever a real x exists, that y is in the range.
Does a hole change the range?
Usually yes. The y-value the curve would have had is missing too. But only if no other x produces that same y, and checking that is the step most working leaves out.

Three rules for the domain, and for the range, solve for x and see which y survive — then check the brackets before you hand it in.