This is a beta course, so its structure, chapters, and examples may continue to change.

Rational Functions

Rational functions connect algebra and analysis, showing up in limits, integration, and partial fractions.

$\mathbb{R}$ （Real numbers）：默认讨论实系数多项式在实数范围内的有理函数。

Definition

Rational function

$f(x) = \dfrac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$ .

符号说明

Symbol	Type	Pronunciation/Note	Meaning in this article
$P(x), Q(x)$	Mathematical symbol	P of x, Q of x	Numerator and denominator polynomials
$\mathbb{R}$	Mathematical symbol	Real numbers	Set of real numbers being discussed

Domain: exclude roots of $Q(x)$ .
Intercepts: $f(0)$ if defined; solve $P(x)=0$ for $x$ -intercepts.
Asymptotes:
- Vertical: $Q(x)=0$ (after simplification).
- Horizontal/oblique: compare degrees of $P$ and $Q$ ; long divide if $\deg P \ge \deg Q$ .
End behavior: leading terms decide the growth and sign at infinity.

$f(x) = \dfrac{1}{x-2}$ : domain $x \neq 2$ ; vertical asymptote $x=2$ ; horizontal asymptote $y=0$ .
$f(x) = \dfrac{x^2-1}{x-1}$ simplifies to $x+1$ except at $x=1$ (removable hole).

Find the domain and vertical asymptote of $f(x) = \dfrac{2x+3}{x^2-4}$ .

Answer and Analysis(3 个标签)

rational functiondomainvertical asymptote

$x^2-4 = (x-2)(x+2)$ ⇒ domain excludes $\pm2$ .
Vertical asymptotes at $x = -2$ and $x = 2$ .

Determine the horizontal or oblique asymptote of $f(x) = \dfrac{x^2 + 1}{x - 1}$ .

Answer and Analysis(3 个标签)

rational functionoblique asymptotepolynomial division

Long divide: $x^2 + 1 = (x-1)(x+1) + 2$ .
So $f(x) = x + 1 + \dfrac{2}{x-1}$ ⇒ oblique asymptote $y = x + 1$ .

符号	类型	读音/说明	在本文中的含义
$P(x), Q(x)$	数学符号	P of x, Q of x	分子与分母多项式
$\deg P$	数学符号	degree of P	多项式的次数
$\mathbb{R}$	数学符号	Real numbers	讨论的数集
$x= a$	数学符号	x equals a	竖直渐近线位置