Rational Function Analyzer

Asymptotes, holes, and behavior of p(x)/q(x)

f(x) = (x² − 1) / (x² − x − 2)

Vertical asymptotes

x = 2

Horizontal asymptote

y = 1

Holes (removable)

(-1, 0.6667)

x-intercepts

x = 1

y-intercept

y = 0.5

Domain

all real numbers except x = 2, x = -1

About this calculator

The Rational Function Analyzer takes a numerator and denominator polynomial p(x)/q(x) and finds the vertical asymptotes, removable holes, horizontal or oblique asymptote, x- and y-intercepts, and domain, then plots the curve with every feature labeled. Useful for precalculus and calculus homework on rational functions.

How to use the Rational Function Analyzer calculator

Enter the values for your problem into the input fields.
Read the result — it updates instantly as you type.
Check the formula and the visual explanation to follow how the answer was found.
Copy the page URL to share the exact calculation.

Common examples

(x²−1)/(x²−x−2) → hole at (−1, 2/3), vertical asymptote x = 2, horizontal asymptote y = 1
1/x → vertical asymptote x = 0, horizontal asymptote y = 0
(x²+1)/x → vertical asymptote x = 0, oblique asymptote y = x
1/(x²−4) → vertical asymptotes x = ±2, horizontal asymptote y = 0
(2x²+3)/(x²+1) → no vertical asymptotes, horizontal asymptote y = 2

Frequently asked questions

How does the Rational Function Analyzer calculator work?

Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.

When would I use the Rational Function Analyzer calculator?

It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.

How accurate are the results?

Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.

Can I share a specific calculation with someone else?

Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.