Definition of Rational Functions
A rational function is a function defined by a rational expression — a ratio of two polynomials, f(x) = P(x)/Q(x). The skills you use to simplify rational expressions carry directly into analyzing rational functions.
Domain Restrictions
Vertical and Horizontal Asymptotes
Holes (Removable Discontinuities)
Want to check your understanding?
Our interaction checks test whether you truly understand a concept — not just whether you can repeat a procedure.
Try an interaction checkGraphing Rational Functions
Using Long Division to Find Asymptotes
When the numerator degree exceeds the denominator degree, use polynomial long division to find oblique asymptotes and simplify the function.
Common Misconceptions
Thinking the function cannot cross a horizontal asymptote
A rational function can cross its horizontal asymptote. The asymptote describes end behavior as x approaches infinity, not behavior for all x values.
Confusing holes with vertical asymptotes
Both involve the denominator being zero, but holes occur when the factor cancels. Always factor first before identifying asymptotes.
Practice Problems
Related Guides
Frequently Asked Questions
What is a rational function?
A rational function is a ratio of two polynomials, written as f(x) = P(x)/Q(x) where Q(x) is not zero. Examples include 1/x, (x+1)/(x-2), and (x²+1)/(x³-x). They are one of the most important function families in algebra and calculus.
How do you find vertical asymptotes?
Set the denominator equal to zero and solve. The x-values where the denominator is zero (and the numerator is not also zero at that point) are vertical asymptotes. If both numerator and denominator are zero, you may have a hole instead.
What is the difference between a hole and a vertical asymptote?
A hole (removable discontinuity) occurs when a factor cancels between the numerator and denominator. A vertical asymptote occurs when the denominator is zero but the factor does not cancel. At a hole, the function is undefined but has a finite limit; at a vertical asymptote, the function approaches infinity.
How do you find horizontal asymptotes?
Compare the degrees of the numerator and denominator. If the numerator degree is less, the horizontal asymptote is y = 0. If the degrees are equal, the asymptote is the ratio of the leading coefficients. If the numerator degree is greater, there is no horizontal asymptote (but there may be an oblique asymptote).
What is an oblique (slant) asymptote?
An oblique asymptote occurs when the numerator degree is exactly one more than the denominator degree. To find it, perform polynomial long division — the quotient (ignoring the remainder) is the equation of the slant asymptote.
Why are rational functions important in calculus?
Rational functions appear frequently in limits, derivatives, and integrals. Techniques like partial fraction decomposition, L'Hôpital's rule, and integration by substitution all involve rational functions. They also model real-world phenomena like rates, concentrations, and inverse-square laws.
About Sense of Study
Sense of Study is a concept-first learning platform that helps students build deep understanding in math, physics, chemistry, statistics, and computational thinking. Our approach maps prerequisite relationships between concepts so students master foundations before moving forward — eliminating the gaps that cause confusion later.
With 800+ interconnected concepts and mastery tracking, we help students and parents see exactly where understanding breaks down and how to fix it.
Start Your Concept Mastery Journey
Explore 800+ interconnected concepts with prerequisite maps, mastery tracking, and interaction checks that build real understanding.