Basic Concepts of Functions

Functions describe how one quantity depends on another. Mastering the definition, how to represent a function, and how to read its domain and range will unlock every later topic in calculus.

What Is a Function?

Definition of a Function

Let $x$ and $y$ be two variables. If every $x$ in some set $D$ is paired with exactly one $y$, then $y$ is called a function of $x$, written $y = f(x)$. Here $x$ is the independent variable, $y$ is the dependent variable, and $D$ is the domain.

Ways to Represent Functions

Analytical (formula) form

• Example: $f(x) = x^2 + 1$
• Pros: precise, easy to compute and analyze.
• Cons: complex or piecewise relationships may need multiple formulas.

Table form

List sample inputs and outputs.

$x$	-1	0	1	2
$f(x) = x^2$	1	0	1	4

Graph form

• Plot $(x, f(x))$ in the plane.
• Great for spotting monotonicity, extrema, intercepts, and symmetry.
• Less precise for exact numeric values.

Domain and Range

Domain and Range

• Domain $D(f)$: all $x$ values that make $f(x)$ meaningful.
• Range (image) $R(f)$: all possible function values $f(x)$ when $x \in D(f)$.

Finding a domain quickly

• Fraction: denominator $\neq 0$ (e.g., $f(x) = \frac{1}{x-2}$ ⇒ $x \neq 2$).
• Even root: radicand $\ge 0$ (e.g., $\sqrt{4-x^2}$ ⇒ $-2 \le x \le 2$).
• Logarithm: argument $> 0$ (e.g., $\ln(x+3)$ ⇒ $x > -3$).
• Combined constraints: satisfy all at once (e.g., $\frac{\sqrt{x-1}}{x-3}$ ⇒ $x \ge 1$ and $x \neq 3$).

Describing domains

• Interval: $(-\infty, 2) \cup (2, +\infty)$
• Set builder: $\{x \mid x \ge 1, x \neq 3\}$
• Inequalities: $x \ge 1$ and $x \neq 3$

Typical ranges

• Power function $x^2$: $[0, +\infty)$
• Exponential $a^x$ ($a>1$): $(0, +\infty)$
• Logarithm $\log_a x$ ($a>1$): $\mathbb{R}$
• Sine and cosine: $[-1, 1]$

Common Classifications

Monotonicity

• Increasing: $x_1 < x_2 \Rightarrow f(x_1) \le f(x_2)$ on an interval.
• Decreasing: $x_1 < x_2 \Rightarrow f(x_1) \ge f(x_2)$ on an interval.

Parity (even/odd)

• Domain must be symmetric about the origin.
• Even: $f(-x) = f(x)$ ⇒ graph symmetric about the $y$-axis.
• Odd: $f(-x) = -f(x)$ ⇒ graph symmetric about the origin.

Periodicity

• If $\exists T \neq 0$ such that $f(x + T) = f(x)$ for all $x$ in the domain, $f$ is periodic. The smallest positive $T$ is the fundamental period.
• Examples: $\sin x$ ($T = 2\pi$), $\tan x$ ($T = \pi$).

Boundedness

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

Building New Functions

• Composite function: if $y = f(u)$ and $u = g(x)$, then $y = f(g(x))$.
• Inverse function: if $f$ is one-to-one on $D$, then $f^{-1}$ exists and satisfies $f(f^{-1}(y)) = y$.
• Piecewise function: different formulas on different intervals (e.g., absolute value).

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练习题

练习 1

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

参考答案

1. $\dfrac{1}{\sqrt{4 - x^2}}$ requires $4 - x^2 > 0$ ⇒ $-2 < x < 2$.
2. $\ln(x)$ requires $x > 0$.
3. Intersection: $(0, 2)$.

练习 2

Determine whether $f(x) = \ln\!\left(\dfrac{1 - x}{1 + x}\right)$ is even, odd, or neither.

参考答案

The domain $(-1, 1)$ is symmetric. Compute $f(-x) = \ln\!\left(\dfrac{1 + x}{1 - x}\right) = -\ln\!\left(\dfrac{1 - x}{1 + x}\right) = -f(x)$.
So $f$ is odd.

练习 3

Give one example each of an increasing, decreasing, even, and periodic function.

参考答案

• Increasing: $f(x) = x$ on $\mathbb{R}$
• Decreasing: $f(x) = -x$ on $\mathbb{R}$
• Even: $f(x) = x^2$ on $\mathbb{R}$
• Periodic: $f(x) = \sin x$ with period $2\pi$

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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	以 $x$ 为自变量的函数
$D(f)$	数学符号	domain of f	函数的定义域
$R(f)$	数学符号	range of f	函数的值域
$\mathbb{R}$	数学符号	Real numbers	全体实数集合
$T$	数学符号	T	周期函数的周期

中英对照

中文术语	英文术语	音标	说明
定义域	domain	/dəʊˈmeɪn/	使函数有意义的自变量取值集合
值域	range	/reɪndʒ/	函数可能取到的输出集合
单调递增	monotonically increasing	/ˌmɒnəʊˈtɒnɪkli ɪnˈkriːsɪŋ/	随自变量增大而不减小
单调递减	monotonically decreasing	/ˌmɒnəʊˈtɒnɪkli dɪˈkriːsɪŋ/	随自变量增大而不增大
偶函数	even function	/ˈiːvn ˈfʌŋkʃən/	满足 $f(-x)=f(x)$ 的函数
奇函数	odd function	/ɒd ˈfʌŋkʃən/	满足 $f(-x)=-f(x)$ 的函数
周期函数	periodic function	/ˌpɪərɪˈɒdɪk ˈfʌŋkʃən/	以固定周期重复的函数
有界函数	bounded function	/ˈbaʊndɪd ˈfʌŋkʃən/	存在统一界 $M$ 满足 $

NextProperties of FunctionsMonotonicity, boundedness, parity, periodicity, extrema, convexity, and inflection points—how to read and use each property.