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

Irrational (Radical) Functions

Radical expressions often encode geometric lengths and appear in limits, derivatives, and integrals.

域的提醒：偶次根号下的表达式必须非负；奇次根可以接受任意实数。

Definition

Irrational (radical) function

$f(x) = \sqrt[n]{g(x)}$ where $n$ is a positive integer and $g(x)$ is another function.

符号说明

Symbol	Type	Pronunciation/Note	Meaning in this article
$\sqrt[n]{g(x)}$	Mathematical symbol	n-th root of g of x	General form of radical function
$\mathbb{R}$	Mathematical symbol	Real numbers	Set of real numbers being discussed

Even $n$ : require $g(x) \ge 0$ for real outputs.
Odd $n$ : domain may be all real numbers.

Typical Shapes

$f(x) = \sqrt{x}$ : domain $[0,+\infty)$ , increasing, concave down.
$f(x) = \sqrt{4 - x^2}$ : domain $[-2,2]$ , upper semicircle.
$f(x) = \sqrt[3]{x}$ : domain $\mathbb{R}$ , odd symmetry.

Techniques

Rationalization: multiply by conjugates to simplify limits or equations.
Domain checks: combine conditions when radicals appear in fractions.
Composition: $\sqrt{g(x)}$ inherits domain constraints from $g$ .

Exercises

Exercise 1

Find the domain of $f(x) = \dfrac{\sqrt{x-1}}{x-3}$ .

Answer and Analysis(3 个标签)

irrational functiondomainradical function

$x-1 \ge 0$ ⇒ $x \ge 1$ ; also $x \ne 3$ .
Domain: $[1,3) \cup (3,+\infty)$ .

Exercise 2

Solve $\sqrt{x+5} = x - 1$ in $\mathbb{R}$ .

Answer and Analysis(3 个标签)

irrational functionradical equationdomain restrictions

Require $x-1 \ge 0$ ⇒ $x \ge 1$ . Square both sides: $x + 5 = (x-1)^2 = x^2 - 2x + 1$
⇒ $x^2 - 3x - 4 = 0$ ⇒ $(x-4)(x+1)=0$ ⇒ candidates $x=4$ or $x=-1$ .
Check domain: only $x=4$ is valid.

总结

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$\sqrt[n]{g(x)}$	数学符号	n-th root of g of x	根式函数的一般形式
$[a,b]$	数学符号	closed interval a b	定义域示例的区间表示
$\mathbb{R}$	数学符号	Real numbers	讨论的数集

中英对照

中文术语	英文术语	音标	说明
无理函数	irrational function	/ɪˈræʃənəl ˈfʌŋkʃən/	含根号的函数形式
根式函数	radical function	/ˈrædɪkəl ˈfʌŋkʃən/	以根号表示的函数
定义域	domain	/dəʊˈmeɪn/	使根号有意义的自变量集合
有效解	valid solution	/ˈvælɪd səˈluːʃən/	满足原方程和定义域的解

PreviousRational FunctionsFunctions built from ratios of polynomials—their domains, asymptotes, and quick analysis steps.

NextFunction ExercisesPractice questions that reinforce domains, properties, and classic constants.