Rational Function
In short: A rational function is a quotient of two polynomials in x, such as (3x – 6)/(x + 2); its graph splits wherever the denominator vanishes and often bends toward asymptote lines.
Algebra 2Functions and GraphsF-IF.C.7d
What is Rational Function?
A rational function takes an input, runs it through a polynomial on top and a polynomial on the bottom, and returns the quotient. Because division by zero is impossible, its graph has gaps, and those gaps shape the curve around them.
Written out, \(f(x) = \frac{p(x)}{q(x)}\), where \(p\) and \(q\) are polynomials and \(q\) is never the zero polynomial. Every real number is allowed except the roots of \(q\). Roots of \(p\) that survive become x-intercepts, while the degrees of \(p\) and \(q\) settle where the graph levels off.
How to spot it
- The formula is a quotient with a polynomial in the denominator, such as (3x – 6)/(x + 2).
- A graph shows two separate branches that approach a vertical dashed line without touching it.
- A story involves a cost or concentration per unit, where the amount shrinks as the quantity grows.
- The directions ask for the domain, intercepts or asymptotes of a rational function given as a fraction in x.
The method
- Find the domain by setting the denominator equal to zero and excluding those x-values.
- Set the numerator equal to zero for x-intercepts, and compute \(f(0)\) for the y-intercept.
- Weigh the degrees to locate the flat guide line; excluded x-values with no cancellation are the vertical guide lines.
- Plot a few extra points on each side of every vertical asymptote and sketch the branches.
Rational Function: solved example
Problem
For \(f(x) = \frac{3x – 6}{x + 2}\), find the domain, the intercepts and both asymptotes.
Solution
The bottom is zero at \(x = -2\), so the domain is all real numbers except \(-2\), and \(x = -2\) is a vertical asymptote because nothing cancels. The top is zero at \(x = 2\), giving the x-intercept \((2, 0)\); also \(f(0) = \frac{-6}{2} = -3\), so the y-intercept is \((0, -3)\).
Top and bottom share degree 1, so the graph levels off at the quotient of the lead coefficients, \(y = \frac{3}{1}\). In summary: domain \(x \neq -2\), vertical asymptote \(x = -2\), horizontal asymptote \(y = 3\).
Try it
Question 1
State the domain of \(g(x) = \frac{x + 1}{x^2 – 25}\).
Show the answer
The bottom is \((x – 5)(x + 5)\), so the domain is all real numbers except \(5\) and \(-5\).
Question 2
Find the horizontal asymptote of \(h(x) = \frac{4x + 7}{x^2 + 1}\).
Show the answer
The denominator has the larger degree, so the asymptote is \(y = 0\).
Rational Function vs. Rational Expression
The expression is only the fraction itself, something you simplify or combine. This page’s topic is the function built from it: a rule with a domain, a graph, intercepts and asymptotes. Two expressions can look different yet define the same function everywhere except at an excluded point.
Keep going
Lessons
Worksheets and quizzes
- Rational Functions and Asymptotes – Worksheet
- Rational Functions and Asymptotes – Quiz
- Rational Expressions – Worksheet
- Rational Expressions – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Functions and Graphs