Rational Functions

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

Definition

Rational function

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

• Domain: all $x$ such that $Q(x) \neq 0$.
• Zeros: solutions of $P(x) = 0$ that are not canceled by $Q(x)$.
• Poles (vertical asymptotes): zeros of $Q(x)$ that are not canceled by $P(x)$.

Quick Analysis Checklist

• 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.

Examples

• $f(x) = \dfrac{1}{x-2}$: 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).

---

练习题

练习 1

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

参考答案

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

练习 2

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

参考答案

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

---

总结

本文出现的符号

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

中英对照

中文术语	英文术语	音标	说明
有理函数	rational function	/ˈræʃənəl ˈfʌŋkʃən/	多项式比值形成的函数
定义域	domain	/dəʊˈmeɪn/	使分母不为零的所有 $x$
渐近线	asymptote	/ˈæsɪmptəʊt/	函数无限接近但不相交的直线
垂直渐近线	vertical asymptote	/ˈvɜːtɪkl ˈæsɪmptəʊt/	分母为零处的竖直线
斜渐近线	oblique asymptote	/əˈbliːk ˈæsɪmptəʊt/	长除法得到的斜线趋势

PreviousThe Golden RatioDefinition, algebra, links to Fibonacci numbers, and quick practice with $\varphi$. NextIrrational (Radical) FunctionsFunctions involving roots—their domains, shapes, and common techniques.