Rational Functions — Mathematics Knowledge System

Overview

A rational function is a ratio of two polynomials, f(x) = p(x)/q(x), with q(x) not identically zero. Unlike polynomials, rational functions can have gaps (holes) and vertical/horizontal/slant asymptotes wherever the denominator vanishes or dominates, making their graphs qualitatively richer and requiring careful analysis of both numerator and denominator behavior.

Intuition

A rational function behaves like ordinary division: wherever the denominator gets close to zero, the fraction's value explodes toward ±∞ (a vertical asymptote), and wherever x itself grows huge, the fraction settles toward whatever ratio the leading terms dictate (a horizontal or slant asymptote). If a factor cancels between numerator and denominator, that's a point where the function is simply undefined — a hole — rather than a genuine explosion.

Interactive Graph

Formal Definition

Definition

A rational function is defined as the quotient of two polynomials, with domain excluding the denominator's zeros:

f(x) = \frac{p(x)}{q(x)}, \quad q(x) \not\equiv 0

General form

\text{dom}(f) = \{x \in \mathbb{R} : q(x) \neq 0\}

The domain excludes every real zero of the denominator

Properties

Vertical asymptote

x=a \text{ is a vertical asymptote if } q(a)=0 \text{ and } p(a) \neq 0

Removable discontinuity (hole)

x=a \text{ is a hole if } (x-a) \text{ is a common factor of both } p(x) \text{ and } q(x)

Horizontal asymptote rules

\deg p < \deg q \Rightarrow y=0;\quad \deg p = \deg q \Rightarrow y = \tfrac{a_n}{b_m};\quad \deg p > \deg q \Rightarrow \text{none (possible slant/curved asymptote)}

Slant asymptote condition

\deg p = \deg q + 1 \Rightarrow \text{slant asymptote, found via polynomial division}

Applications

Lens and mirror equations, and the relationship between resistance in parallel circuits, are naturally expressed as rational functions.

Worked Examples

Factor both numerator and denominator.
$f(x) = \frac{(x-2)(x+2)}{(x-2)(x+1)}$
(x-2) cancels, giving a hole at x=2; the remaining factor (x+1) in the denominator gives a vertical asymptote at x=-1.
$f(x) = \frac{x+2}{x+1},\ x \neq 2$

Answer: Domain: x≠2, x≠-1. Hole at x=2. Vertical asymptote at x=-1.

Practice Problems

Difficulty 4/10

Find the horizontal asymptote of f(x) = (3x²+1)/(2x²-x+5).

Difficulty 5/10

Find all vertical asymptotes and holes of f(x) = (x-3)/(x²-9).

Common Mistakes

Common Mistake

Treating every zero of the denominator as a vertical asymptote.

If the same factor also appears in the numerator and cancels, that zero is a HOLE (removable discontinuity), not a vertical asymptote. Only zeros of the denominator that survive after cancellation give true vertical asymptotes.

Common Mistake

Assuming a rational function always has a horizontal asymptote.

A horizontal asymptote only exists when deg(numerator) ≤ deg(denominator). If the numerator's degree exceeds the denominator's by exactly 1, there's a slant asymptote instead; if by more than 1, there's neither, and the function grows unboundedly like a curve.

Summary

A rational function f(x)=p(x)/q(x) is undefined wherever q(x)=0.
A denominator zero gives a vertical asymptote unless the same factor cancels with the numerator, in which case it's a hole instead.
Horizontal asymptotes compare degrees: equal degrees give y=aₙ/bₘ; numerator degree less gives y=0; numerator degree greater gives no horizontal asymptote.
When numerator degree is exactly one more than denominator degree, polynomial division reveals a slant (oblique) asymptote.
Rational functions model division-based relationships: transfer functions, average cost, parallel resistances, and lens equations.

References

BookSullivan, M. Algebra and Trigonometry, 10th ed. Ch. 5 — Polynomial and Rational Functions.
WebsiteWikipedia — Rational function