Properties of Functions

Understanding properties lets you sketch graphs, solve inequalities, and invert functions with confidence.

Monotonicity

Monotonicity on an interval

Let $f(x)$ be defined on interval $I$.

• Strictly increasing: $x_1 < x_2 \Rightarrow f(x_1) < f(x_2)$ for all $x_1, x_2 \in I$.
• Strictly decreasing: $x_1 < x_2 \Rightarrow f(x_1) > f(x_2)$ for all $x_1, x_2 \in I$.

• Geometric view: increasing curves go up left-to-right; decreasing go down.
• Derivative test (when differentiable): $f'(x) > 0$ ⇒ strictly increasing; $f'(x) < 0$ ⇒ strictly decreasing.

Boundedness

• $f$ is bounded on $I$ if $\exists M$ with $|f(x)| \le M$ for all $x \in I$.
• Example: $\sin x$ is bounded by 1 on $\mathbb{R}$; $x^2$ is bounded on $[-1,1]$ by 1.

Extrema

• Local maximum at $x_0$: there is a neighborhood where $f(x) \le f(x_0)$.
• Local minimum: $f(x) \ge f(x_0)$ nearby.
• Typical derivative test: $f'(x_0) = 0$ and sign of $f'$ changes from + to – (max) or – to + (min).

Parity

• Domain must be symmetric about the origin.
• Even: $f(-x) = f(x)$ ⇒ symmetric about the $y$-axis (e.g., $x^2$, $\cos x$).
• Odd: $f(-x) = -f(x)$ ⇒ symmetric about the origin (e.g., $x^3$, $\sin x$).

Periodicity

• $f$ is periodic if $\exists T \neq 0$ with $f(x+T) = f(x)$ for all $x$.
• Smallest positive $T$ is the fundamental period.
• Examples: $\sin x$ period $2\pi$; $\tan x$ period $\pi$.

Convexity and Inflection

• Convex (concave up) on $I$: the line segment between any two points on the graph lies above the curve; analytically $f''(x) \ge 0$ (when it exists).
• Concave (concave down): $f''(x) \le 0$.
• Inflection point: where curvature changes sign (typically $f''$ changes sign).

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练习题

练习 1

Determine where $f(x) = x^3 - 3x + 1$ is increasing or decreasing.

参考答案

$f'(x) = 3x^2 - 3 = 3(x-1)(x+1)$

• $x < -1$: $f'(x) > 0$ ⇒ increasing
• $-1 < x < 1$: $f'(x) < 0$ ⇒ decreasing
• $x > 1$: $f'(x) > 0$ ⇒ increasing

练习 2

Is $f(x) = \ln\!\left(\dfrac{1+x}{1-x}\right)$ odd or even on $(-1,1)$?

参考答案

Domain is symmetric. $f(-x) = \ln\!\left(\dfrac{1 - x}{1 + x}\right) = -\ln\!\left(\dfrac{1 + x}{1 - x}\right) = -f(x)$, so $f$ is odd.

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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
$f(x)$	数学符号	f of x	研究的函数
$I$	数学符号	interval	讨论性质的区间
$f'(x)$	数学符号	f prime of x	函数的一阶导数
$f''(x)$	数学符号	f double prime of x	函数的二阶导数
$T$	数学符号	T	周期
$\mathbb{R}$	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
单调性	monotonicity	/ˌmɒnəʊtəˈnɪsɪti/	函数随自变量变化的增减趋势
有界性	boundedness	/ˈbaʊndɪdnəs/	函数值被统一常数所约束
极大值	local maximum	/ˈləʊkl ˈmæksɪməm/	在邻域内函数值最大的点
极小值	local minimum	/ˈləʊkl ˈmɪnɪməm/	在邻域内函数值最小的点
奇函数	odd function	/ɒd ˈfʌŋkʃən/	满足 $f(-x)=-f(x)$ 的函数
偶函数	even function	/ˈiːvn ˈfʌŋkʃən/	满足 $f(-x)=f(x)$ 的函数
周期性	periodicity	/ˌpɪərɪəˈdɪsəti/	函数值按固定周期重复
凸性	convexity	/kɒnˈvɛksəti/	函数曲线向上弯曲（$f''\ge0$）
拐点	inflection point	/ɪnˈflɛkʃən pɔɪnt/	曲率符号发生变化的点

Chapters

• 函数的有界性
• 函数的单调性
• 函数的周期性
• 函数的奇偶性
• 函数的极值
• 极值的判定方法
• 函数的凹凸性
• 函数的拐点

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