Rational Function¶

Definition¶

A rational function is the quotient of two polynomials when the denominator is not always equal to zero.

Interpretation¶

A rational function divides one polynomial by another, $f(x)=P(x)/Q(x)$ $P (x) / Q (x)$ . It behaves like a polynomial wherever the denominator is nonzero; its special features come from the zeros of $Q$ and from comparing the degrees of $P$ and $Q$ .

The natural domain is the set of $x$ with $Q(x)\neq0$ $0$ . Because $Q$ is not the zero polynomial, it has at most $\deg Q$ zeros, so only finitely many points are left out. Near a zero $c$ of $Q$ there are two cases. If $P(c)\neq0$ $0$ , the values $|f(x)|$ become arbitrarily large as $x$ approaches $c$ , as for $1/x$ near $0$ . If $P(c)=0$ as well, the factor $x-c$ can be cancelled from numerator and denominator, and after cancelling the function may behave perfectly well near $c$ : $\frac{x^2-1}{x-1}=x+1$ for every $x\neq1$ $1$ .

For large $|x|$ only the leading terms matter. If $\deg P<\deg Q$ $de g Q$ , the values get close to $0$ ; if the degrees are equal, they get close to the ratio of the leading coefficients; if $\deg P>\deg Q$ $de g Q$ , they grow without bound. In the third Example, $\deg P=3>\deg Q=2$ $2$ , and division gives

f(x)=\frac{x-1}{3}+\frac{11x-14}{3(3x^2+1)},

where the last term is small for large $|x|$ , so the graph runs close to the line $y=(x-1)/3$ $1) /3$ .

A common slip is to simplify first and read off the domain afterwards. The functions $\frac{x^2-1}{x-1}$ and $x+1$ agree everywhere except at $x=1$ $1$ , where only the second is defined. Later these patterns are made precise with limits, as vertical and horizontal asymptotes.

Key Results¶

Rational functions are undefined where the denominator vanishes.
The behaviour as $x\to\pm\infty\text{,}$ including any horizontal asymptote, is determined by the degrees and leading coefficients of the two polynomials.
A rational function is continuous at every point of its domain; the zeros of the denominator are excluded from the domain, not points of discontinuity.

Examples¶

$f(x)=\dfrac{1}{x}$ is a rational function.
$f(x)=\dfrac{x^2-1}{x-1}$ is undefined at $x=1\text{.}$
$f(x)=\dfrac{x^3-x^2+4x-5}{3x^2+1}$ is defined everywhere.

The graph of a rational function with a denominator 3x squared plus 1 that never vanishes, an unbroken curve over the whole real line.

The denominator $3x^2+1$ never vanishes, so the graph is one unbroken curve over all of $\mathbb{R}$ .

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 4

Symbols: infinity, real numbers

Last updated: 20 October 2026 03:02