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 function domain radical 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 function radical equation domain 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.

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