Function Exercises

Tackle these to solidify the ideas from the course.

提示：先检查定义域，再考虑性质（单调、奇偶、周期）和变换。

练习 1

Find the domain of $f(x) = \dfrac{1}{x-1} + \sqrt{4 - x^2}$ .

参考答案

Require $x \ne 1$ and $4 - x^2 \ge 0$ ⇒ $-2 \le x \le 2$ .
Domain: $[-2,1) \cup (1,2]$ .

Determine whether $f(x) = x^3 - x$ is odd, even, or neither. State its monotonicity.

参考答案

$f(-x) = -x^3 + x = -f(x)$ ⇒ odd.
$f'(x) = 3x^2 - 1$ : decreasing on $\left(-\tfrac{1}{\sqrt{3}}, \tfrac{1}{\sqrt{3}}\right)$ , increasing elsewhere.

Compute $\displaystyle \lim_{x\to 0} \dfrac{e^x - 1}{x}$ .

参考答案

Standard limit for the exponential: $\lim_{x\to 0}\dfrac{e^x - 1}{x} = 1$ .

符号	类型	读音/说明	在本文中的含义
$f(x)$	数学符号	f of x	练习中的函数
$\lim$	数学符号	limit	极限运算符
$e^x$	数学符号	e to the x	指数函数

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Exploring Functions in Advanced Mathematics
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Functions are a core idea of advanced mathematics. This course walks through foundational concepts, key properties, and classic constants so you can read, reason, and compute with confidence.
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