Domain of a Rational Function Calculator

DOMAIN

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Enter the function, then press = to find its domain.

Every excluded value, and what it does to the graph

Numerator (the top)

Denominator (the bottom)

The domain comes from the denominator as it was written. A factor that cancels still excludes its value — it leaves a hole in the graph rather than an asymptote, and it is the one people miss.

Examples — tap one and it solves

How the domain is found, step by step

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Take f(x) = (x² − 4)/(x − 2). Cancel the (x − 2) and you are left with x + 2, which is defined everywhere — so it is easy to write down “all real numbers” and move on.

But the function you were handed is undefined at x = 2, and cancelling does not give it a value it never had. The domain is x ≠ 2, and the graph has a single missing point there rather than an asymptote. This domain of a rational function calculator takes the domain from the denominator as written, and tells you which excluded values are holes and which are asymptotes.

How to Find the Domain of a Rational Function

Set the bottom equal to zero and solve it. Everything else is in. That is the whole method, and it takes about ten seconds once the denominator is factored — which is why the marks are lost somewhere else.

They are lost at the next step, deciding what each excluded value does. A factor that also sits on top cancels, and the graph gets a hole: one point missing from a curve that is otherwise perfectly ordinary. A factor that does not cancel gives a vertical asymptote, where the function runs off to infinity. Both are out of the domain, but they look nothing alike on a graph, and questions ask about both.

The calculator factors the denominator, lists every value it excludes, and labels each one. Where a factor appears more than once it compares the powers — (x−1)³ underneath with (x−1)² on top still leaves one behind, so that is an asymptote, not a hole. Excluded values that are irrational, like the ±√2 from x² − 2, are kept exactly rather than rounded or quietly dropped.

How to Use This Domain Calculator

Put the numerator in the top box and the denominator in the bottom, then press =. The bottom can be factored, as (x−1)(x+2), or expanded as x²+x−2 — it is factored for you either way.

The domain appears in interval notation with the short form beneath it, and one tap opens the table of excluded values with what each one does to the graph. Set-builder notation is there too, since different classes ask for different forms. Free to use online, with no account.

Frequently Asked Questions

How do you find the domain of a rational function?

Set the denominator equal to zero and solve. Every value you get is excluded, and everything else is in the domain. Use the denominator you were given, not the one left after cancelling.

Does a factor that cancels still affect the domain?

Yes. It leaves a hole in the graph rather than a vertical asymptote, but the value is still excluded. This is the part most often lost, because the simplified function looks perfectly well behaved there.

What is the difference between a hole and a vertical asymptote?

Both are excluded from the domain. A hole is a single missing point on an otherwise ordinary curve, and happens when the factor cancels against the numerator. An asymptote is where the function runs off to infinity, and happens when nothing cancels it.

Factor the bottom, set it to zero, and take every answer — including the ones that cancel. That is the domain.