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

Special Function Forms

Not all functions are given by one simple formula. These five special forms appear everywhere in analysis and applications.

符号说明： $f^{-1}$ 表示反函数； $F(x, y)=0$ 表示隐函数方程； $(x(t), y(t))$ 表示参数方程。

Composite Functions

Composite function

If $u = g(x)$ and $y = f(u)$ , then $y = f(g(x))$ is the composite of $f$ and $g$ .
Domain: all $x$ such that $x \in D(g)$ and $g(x) \in D(f)$ .

Example: $f(u) = e^u$ , $g(x) = x^2 + 1$ ⇒ $f(g(x)) = e^{x^2+1}$ .

Inverse Functions

Inverse function

If $f$ is one-to-one on $D$ , the inverse $f^{-1}$ satisfies $f(f^{-1}(y)) = y$ and $f^{-1}(f(x)) = x$ .

Graphs of $f$ and $f^{-1}$ are symmetric about $y = x$ .
Monotonicity is preserved: if $f$ is strictly increasing, so is $f^{-1}$ .

Piecewise Functions

Different expressions on different intervals.
Pay attention to continuity at the breakpoints.
Example: absolute value $f(x) = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}$ .

Implicit Functions

Relation given by $F(x, y) = 0$ instead of $y = f(x)$ explicitly.
Under mild conditions, the implicit function theorem guarantees a local function $y = \varphi(x)$ .
Example: circle $x^2 + y^2 - 1 = 0$ ⇒ $y = \pm\sqrt{1 - x^2}$ .

Parametric Functions

Both variables expressed via a parameter $t$ : $(x(t), y(t))$ .
Great for curves that are hard to describe explicitly (cycloids, Lissajous curves).
Tangent slope: $\dfrac{dy}{dx} = \dfrac{y'(t)}{x'(t)}$ when $x'(t) \neq 0$ .

练习题

练习 1

Given $f(x) = \sqrt{2x+3}$ and $g(x) = x^2 - 1$ , write $f(g(x))$ and find its domain.

参考答案

$f(g(x)) = \sqrt{2(x^2 - 1) + 3} = \sqrt{2x^2 + 1}$ .
Need $2x^2 + 1 \ge 0$ , always true ⇒ domain $\mathbb{R}$ .

练习 2

Find the inverse of $y = e^{2x} + 1$ and its domain.

参考答案

$y - 1 = e^{2x} \Rightarrow \ln(y-1) = 2x \Rightarrow x = \tfrac{1}{2}\ln(y-1)$ .
Swap $x, y$ : $y = \tfrac{1}{2}\ln(x-1)$ with domain $(1, +\infty)$ .

练习 3

For the parametric curve $x = \cos t$ , $y = \sin t$ , compute $\dfrac{dy}{dx}$ .

参考答案

$x'(t) = -\sin t$ , $y'(t) = \cos t$ ⇒ $\dfrac{dy}{dx} = \dfrac{\cos t}{-\sin t} = -\cot t$ (when $\sin t \neq 0$ ).

总结

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$f(g(x))$	数学符号	f of g of x	复合函数
$f^{-1}(x)$	数学符号	f inverse of x	反函数
$F(x, y)=0$	数学符号	F of x y equals zero	隐函数关系
$(x(t), y(t))$	数学符号	x of t, y of t	参数方程
$\dfrac{dy}{dx}$	数学符号	dy over dx	斜率或导数
$\mathbb{R}$	数学符号	Real numbers	全体实数集合

中英对照

中文术语	英文术语	音标	说明
复合函数	composite function	/kəmˈpɒzɪt ˈfʌŋkʃən/	通过嵌套得到的新函数
反函数	inverse function	/ɪnˈvɜːs ˈfʌŋkʃən/	将输出映回输入的函数
分段函数	piecewise function	/ˈpiːswaɪz ˈfʌŋkʃən/	不同区间使用不同表达式
隐函数	implicit function	/ɪmˈplɪsɪt ˈfʌŋkʃən/	由方程关系确定的函数
参数方程	parametric equation	/ˌpærəˈmɛtrɪk ɪˈkweɪʒən/	用参数描述曲线的方程
周期参数	parameter	/pəˈræmɪtər/	描述曲线位置的变量

Chapters

Previous Function Inflection Points

Next Composite Functions Learn the definition, properties, and methods of composite functions, master the method of determining the domain of composite functions.

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Exploring Functions in Advanced Mathematics
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