Special Function Forms

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

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 $x'(t) \neq 0$.

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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 $\sin t \neq 0$).

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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
$f(g(x))$	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

• 复合函数
• 反函数
• 分段函数
• 隐函数
• 参数函数

PreviousProperties of FunctionsMonotonicity, boundedness, parity, periodicity, extrema, convexity, and inflection points—how to read and use each property. NextElementary Functions at a GlancePower, exponential, logarithmic, trigonometric, and inverse trigonometric functions—key shapes, domains, and quick properties.