Solve domains, holes, asymptotes, and intercepts quickly. Enter polynomial coefficients, test values, and review steps. Save clean outputs for lessons, assignments, revision, and reference.
| Numerator coefficients | Denominator coefficients | x value | Simplified function | Hole | Vertical asymptote | f(x) |
|---|---|---|---|---|---|---|
| 1, 0, -1 | 1, -3, 2 | 3 | (x + 1) / (x - 2) | (1, -2) | x = 2 | 4 |
| 2, 5, -3 | 1, -1 | 2 | (2x² + 5x - 3) / (x - 1) | None | x = 1 | 15 |
General form: f(x) = N(x) / D(x), where D(x) ≠ 0.
Function value: Substitute x into the numerator and denominator, then divide.
Domain restriction: Exclude real x values that make D(x) = 0.
x-intercepts: Solve the simplified numerator equal to zero.
y-intercept: Evaluate f(0) when the denominator at zero is not zero.
Vertical asymptotes: Real zeros of the simplified denominator.
Horizontal asymptote: Compare numerator and denominator degrees.
Oblique or polynomial asymptote: Use polynomial long division when the numerator degree is greater.
Derivative: f′(x) = [N′(x)D(x) − N(x)D′(x)] / [D(x)]².
A rational functions calculator helps you study expressions written as one polynomial divided by another. This tool accepts numerator and denominator coefficients. It builds the function and checks the denominator for restricted values. It also evaluates the function at a chosen x value. Students use this kind of algebra tool for homework, revision, and classroom checking.
Rational functions often contain features that are easy to miss during manual work. This calculator highlights domain restrictions, x-intercepts, y-intercepts, vertical asymptotes, and end behavior. It also looks for removable discontinuities. Those discontinuities appear as holes after common real factors cancel. The simplified form can make the graph easier to read. That is useful during polynomial analysis and graph interpretation.
Coefficient entry is fast and flexible. You can test simple linear examples or higher degree expressions with the same form. Entering coefficients from highest degree to constant keeps the input clean. It also mirrors the way many algebra systems process polynomials. Once submitted, the calculator runs evaluation, derivative support, and polynomial division. That creates a fuller rational function report in one place.
This page is useful for algebra students, teachers, tutors, and exam learners. It supports rational expression practice, function analysis, and asymptote checking. The export options also help when you need a saved worksheet result. You can compare trial functions, review solutions, and prepare examples for lessons. In short, the calculator turns coefficient data into practical algebra insight with less effort and fewer mistakes.
A rational function is a quotient of two polynomials. The denominator cannot equal zero. Because of that restriction, rational functions may have holes, asymptotes, and excluded x values.
Enter them from highest degree to constant term. For x² − 1, type 1,0,-1. For x³ + 2x, type 1,0,2,0.
The simplified function shows the expression after detected real common factors cancel. This can reveal holes and make intercepts or asymptotes easier to interpret.
A hole is a removable discontinuity. It happens when the numerator and denominator share a common factor. The factor cancels, but the original function still excludes that x value.
Vertical asymptotes come from real zeros of the simplified denominator. If a denominator zero cancels with a numerator zero, it becomes a hole instead of an asymptote.
f(x) is undefined whenever the denominator becomes zero at the chosen x value. That x sits outside the domain of the original rational function.
They depend on degree comparison. If the numerator degree is smaller, y = 0. If the degrees match, divide the leading coefficients.
Export helps you save results for assignments, tutoring notes, worked examples, and revision sheets. It is also useful when sharing outputs with classmates or students.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.