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

Properties of Functions

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

$\mathbb{R}$ （双线体 R，Real numbers）：表示全体实数集合，本章默认在实数范围讨论函数性质。

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.

$f$ 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.

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).

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., $x^3$ , $\sin x$ ).

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).

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

参考答案

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

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.

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