How to use this calculator

A rational function is a ratio of polynomials. Values making the original denominator zero are excluded even when a common factor cancels. Cancellation can create a removable hole; a remaining denominator zero gives a vertical asymptote. The calculator examines multiplicities rather than treating all denominator zeros alike.

The method

R(x) = N(x) / D(x), with D(x) not equal to 0

Enter the values in the labeled fields, check the units or selected mode, and choose Calculate. The result includes the applicable working and a breakdown or visual when useful. Change an input and calculate again to update the result.

WORKED EXAMPLE

Numerator coefficients: 1, 0, -1 Denominator coefficients: 1, -1 Graph minimum x: -5 Graph maximum x: 5

Domain of the rational function: All real x except 1

(x^2 - 1) / (x - 1)

Scope and limitations

Numerator and denominator degrees are limited to 2. Numerical tolerances are used to compare roots. The graph may clip extreme values near asymptotes; use the feature table for exact interpretation.

Common questions

Does cancelling a factor restore the missing x-value?

No. The original expression is still undefined there. A cancelled factor can produce a removable hole.

How do I enter x^2 - 1?

Enter 1, 0, -1. The zero is necessary to preserve the missing x term.

References & further reading

External references provide background, not an endorsement of this implementation. See our methodology for numerical conventions and limitations.