Irrational (Radical) Functions

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

Definition

Irrational (radical) function

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

• 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 $g$.

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练习题

练习 1

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

参考答案

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

练习 2

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

参考答案

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.

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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

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