Rational function calculator
Type a fraction of two polynomials, such as (x^2 − 1) / (x^2 − x − 2), and get its domain, zeros, vertical asymptotes, holes, the asymptote for large x and the y-intercept.
A ratio of two polynomials in x, for example (2x + 1) / (x^2 − 4).
Fill in the fields; the answer appears here straight away.
How it's worked out
How it works
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Enter the values
Type the numbers or the formula. Decimals may use a point or a comma. No idea? Click Fill in an example.
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Instant answer
The answer appears as you type, with the most important intermediate values.
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See the working
Under "How it's worked out" you see the steps, handy for checking your own calculation.
What it finds
- Domain: all x except the zeros of the denominator.
- Zeros: where the numerator is 0 and the denominator isn't.
- Vertical asymptotes: zeros of the denominator where the numerator isn't 0.
- Holes: values where numerator and denominator are both 0, so a factor cancels.
- Horizontal or oblique asymptote: depends on the degrees of numerator and denominator.
In the example, (x² − 1) / (x² − x − 2) = (x − 1)(x + 1) / ((x − 2)(x + 1)): there is a hole at x = −1, a vertical asymptote at x = 2 and a horizontal asymptote y = 1.
Degrees and asymptotes
If the numerator's degree is lower, the asymptote is y = 0; if equal, y = the ratio of the leading coefficients; one higher gives an oblique asymptote. Only polynomials are allowed, so no sin, roots or powers of e.
Frequently asked questions
Is the rational function calculator free?
Yes. The rational function calculator is completely free, with no account, sign-up or usage limit.
How accurate is the result?
The tool calculates with your browser's number precision. For important calculations, always double-check what you entered.
Does it work on my phone?
Yes. The tool works in any modern browser on a phone, tablet or computer.