Functions: Concepts and Basic Properties | 函数的概念与基本性质

📚 Functions: Concepts and Basic Properties | 函数的概念与基本性质

A function is one of the most fundamental ideas in mathematics, describing a precise relationship between two sets of quantities. In A-level mathematics, mastering the concept of a function and its properties is essential for solving problems across algebra, calculus, and even applied topics such as mechanics and statistics.

函数是数学中最基本的概念之一，描述了两个量之间精确的对应关系。在 A-level 数学中，掌握函数的概念及其基本性质，是解决代数、微积分以及力学、统计学等应用问题的关键基础。

1. What is a Function? | 什么是函数

Formally, a function f from a set A to a set B is a rule that assigns to every element x in A exactly one element f(x) in B. The set A is called the domain, and the set B is called the codomain. The most important condition is that each input has only one output; however, different inputs may share the same output.

严格地说，从集合 A 到集合 B 的函数 f 是一个规则，它把 A 中的每一个元素 x 恰好对应到 B 中的一个元素 f(x)。集合 A 称为定义域，集合 B 称为陪域。最关键的条件是：每个输入只能有一个输出；但不同的输入可以有相同的输出。

For example, the relation y = x² is a function because for every x, there is only one y. However, the relation x² + y² = 1 is not a function, because for a single x (except x = ±1) there are two possible y values.

例如，关系 y = x² 是一个函数，因为对于每一个 x，只有一个 y。而关系 x² + y² = 1 不是函数，因为对于同一个 x（除 x = ±1 外），会出现两个不同的 y 值。

2. Domain and Range | 定义域与值域

The domain of a function is the complete set of possible input values. The range is the set of all outputs that the function actually produces. Note that the range is a subset of the codomain, and sometimes it is much smaller than the codomain.

定义域是函数所有可能输入值的集合。值域是函数实际产生的所有输出值的集合。注意，值域是陪域的子集，有时比陪域小得多。

For f(x) = 1/x, the domain is all real numbers except 0, and the range is also all real numbers except 0.

对于 f(x) = 1/x，定义域是所有非零实数，值域也是所有非零实数。
For f(x) = √x (real-valued), the domain is x ≥ 0 and the range is y ≥ 0.

对于实值函数 f(x) = √x，定义域是 x ≥ 0，值域是 y ≥ 0。
For f(x) = x², the domain is all real numbers, but the range is y ≥ 0.

对于 f(x) = x²，定义域是全体实数，但值域是 y ≥ 0。

In A-level problems, you must always consider the domain before simplifying a function, because operations like division by zero or taking the square root of a negative number restrict the domain.

在 A-level 题目中，化简直前必须先考虑定义域，因为除零或对负数开平方等运算会限制定义域。

3. Function Notation and Evaluation | 函数记号与求值

A function is usually written as f(x) or y = f(x). To evaluate a function at a specific value, replace x with that value and simplify. For instance, if f(x) = 2x + 3, then f(4) = 2(4) + 3 = 11.

函数通常写成 f(x) 或 y = f(x)。求函数在某一点的值时，只需用该值替换 x 并化简。例如，若 f(x) = 2x + 3，则 f(4) = 2(4) + 3 = 11。

You may also encounter expressions like f(x + h), which is essential in calculus for the definition of the derivative. For f(x) = x², we have:

你还会遇到像 f(x + h) 这样的表达式，这在微积分中定义导数时十分重要。对于 f(x) = x²，有：

f(x + h) = (x + h)² = x² + 2xh + h²

Then the difference quotient [f(x + h) − f(x)] / h equals 2x + h, and taking the limit as h → 0 gives the derivative.

于是差商 [f(x + h) − f(x)] / h 等于 2x + h，令 h → 0 取极限便得到导数。

4. Composite Functions | 复合函数

A composite function is formed when the output of one function becomes the input of another. The composite of f and g is written as (f ∘ g)(x) = f(g(x)), meaning “apply g first, then apply f to the result.”

复合函数是指把一个函数的输出作为另一个函数的输入。f 和 g 的复合记为 (f ∘ g)(x) = f(g(x))，意思是”先算 g，再把结果代入 f”。

For example, if f(x) = 2x + 1 and g(x) = x², then:

例如，若 f(x) = 2x + 1，g(x) = x²，则：

(f ∘ g)(x) = f(x²) = 2x² + 1

(g ∘ f)(x) = g(2x + 1) = (2x + 1)² = 4x² + 4x + 1

Notice that f ∘ g and g ∘ f are generally different. The domain of the composite function f(g(x)) is the set of x in the domain of g such that g(x) lies in the domain of f.

注意 f ∘ g 与 g ∘ f 通常不同。复合函数 f(g(x)) 的定义域是 g 定义域中使 g(x) 落在 f 定义域内的那些 x 的集合。

5. Inverse Functions | 反函数

An inverse function reverses the action of a function. If f(a) = b, then f⁻¹(b) = a. For the inverse to exist, the original function must be one-to-one (injective), meaning that no two different inputs give the same output.

反函数是原函数作用的逆运算。若 f(a) = b，则 f⁻¹(b) = a。要存在反函数，原函数必须是单射（一一对应），即不能有两个不同的输入产生相同的输出。

To find the inverse of a function given by an equation, follow these steps:

已知函数方程求反函数的步骤如下：

Write the function as y = f(x).

把函数写成 y = f(x)。
Swap x and y in the equation.

交换方程中的 x 和 y。
Solve the equation for y. The result is y = f⁻¹(x).

解出 y，得到 y = f⁻¹(x)。

For f(x) = 3x − 2, write y = 3x − 2, swap to x = 3y − 2, and solve: y = (x + 2)/3. Thus f⁻¹(x) = (x + 2)/3.

以 f(x) = 3x − 2 为例，写成 y = 3x − 2，交换得 x = 3y − 2，解得 y = (x + 2)/3。所以 f⁻¹(x) = (x + 2)/3。

The graphs of a function and its inverse are reflections of each other across the line y = x. Also, the domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.

函数与其反函数的图像关于直线 y = x 对称。此外，f⁻¹ 的定义域是 f 的值域，f⁻¹ 的值域是 f 的定义域。

6. Graphs of Functions | 函数图像

The graph of a function y = f(x) is the set of all points (x, f(x)) in the coordinate plane. A vertical line test can determine whether a curve is a function: if any vertical line intersects the curve more than once, the curve is not the graph of a function.

函数 y = f(x) 的图像是坐标平面内所有点 (x, f(x)) 的集合。用铅垂线检验可以判断一条曲线是否为函数图像：若任意一条竖直线与曲线交于多于一个点，则该曲线不是函数图像。

Key features of a function graph include intercepts, turning points, asymptotes, and intervals where the function is positive or negative. For example, the graph of f(x) = x² − 4 has x-intercepts at x = −2 and x = 2, and a y-intercept at y = −4.

函数图像的重要特征包括截距、极值点、渐近线以及函数取正或取负的区间。例如，f(x) = x² − 4 的图像在 x = −2 和 x = 2 处有 x 截距，在 y = −4 处有 y 截距。

When sketching graphs, always consider the behaviour as x → ∞ and x → −∞. For rational functions, vertical asymptotes occur where the denominator is zero, and horizontal asymptotes are determined by the degrees of the numerator and denominator.

画图时必须考虑 x → ∞ 和 x → −∞ 时函数的行为。对于有理函数，竖渐近线出现在分母为零处，水平渐近线由分子与分母的次数决定。

7. Even and Odd Functions | 奇偶性

Even and odd functions describe symmetry of a graph. A function is even if f(−x) = f(x) for all x in its domain; its graph is symmetric with respect to the y-axis. A function is odd if f(−x) = −f(x) for all x; its graph is symmetric with respect to the origin.

奇偶性描述函数图像的对称性。若对所有定义域内的 x，都有 f(−x) = f(x)，则函数是偶函数，其图像关于 y 轴对称。若 f(−x) = −f(x)，则函数是奇函数，其图像关于原点对称。

Examples:

举例：

f(x) = x² is even because (−x)² = x².

f(x) = x² 是偶函数，因为 (−x)² = x²。
f(x) = x³ is odd because (−x)³ = −x³.

f(x) = x³ 是奇函数，因为 (−x)³ = −x³。
f(x) = x³ + x is odd, while f(x) = x² + 1 is even.

f(x) = x³ + x 是奇函数，而 f(x) = x² + 1 是偶函数。

Many functions are neither even nor odd, such as f(x) = x² + x. In integration, these properties can simplify definite integrals over symmetric intervals:

许多函数既不是偶函数也不是奇函数，例如 f(x) = x² + x。在积分中，这些性质可以简化对称区间上的定积分：

∫₋ₐ^ₐ f(x) dx = 2∫₀^ₐ f(x) dx if f is even; = 0 if f is odd

8. Monotonic Functions | 单调性

A function is increasing on an interval if x₁ < x₂ implies f(x₁) ≤ f(x₂). It is strictly increasing if f(x₁) < f(x₂). Similarly, a function is decreasing if x₁ < x₂ implies f(x₁) ≥ f(x₂), and strictly decreasing if the inequality is strict.

函数在某个区间上递增，是指 x₁ < x₂ 时总有 f(x₁) ≤ f(x₂)。若 f(x₁) < f(x₂)，则称为严格递增。类似地，若 x₁ < x₂ 时 f(x₁) ≥ f(x₂)，则函数在该区间上递减；若不等号严格成立，则为严格递减。

The derivative is the main tool for studying monotonicity. If f'(x) > 0 on an interval, the function is increasing there; if f'(x) < 0, it is decreasing. For example, f(x) = x² is decreasing for x < 0 and increasing for x > 0.

导数是研究单调性的主要工具。若在某个区间上 f'(x) > 0，则函数在该区间上递增；若 f'(x) < 0，则递减。例如 f(x) = x² 在 x < 0 时递减，在 x > 0 时递增。

A strictly monotonic function is always one-to-one, so it has an inverse. This is why we restrict the domain of f(x) = x² to x ≥ 0 when defining its inverse f⁻¹(x) = √x.

严格单调的函数一定是一一对应的，因此存在反函数。这就是为什么在定义 f(x) = x² 的反函数 f⁻¹(x) = √x 时，要把定义域限制为 x ≥ 0。

9. Bounded Functions | 有界性

A function is bounded above if there exists a real number M such that f(x) ≤ M for all x in its domain. It is bounded below if there exists a real number m such that f(x) ≥ m for all x. A function is bounded if it is bounded both above and below.

若存在实数 M，使得定义域内所有 x 都有 f(x) ≤ M，则函数上有界。若存在实数 m，使得所有 x 都有 f(x) ≥ m，则函数下有界。若函数既有上界又有下界，则称其为有界函数。

For example, f(x) = sin x is bounded because −1 ≤ sin x ≤ 1. The function f(x) = x² is bounded below by 0 but not bounded above. The function f(x) = 1/x with domain x > 0 is bounded below by 0 but not bounded above.

例如 f(x) = sin x 是有界函数，因为 −1 ≤ sin x ≤ 1。f(x) = x² 下有界（下界为 0）但无上界。定义域为 x > 0 的 f(x) = 1/x 下有界（下界为 0）但无上界。

Boundedness is important in analysis and calculus, especially for the existence of maxima and minima on closed intervals under the extreme value theorem.

有界性在数学分析和微积分中非常重要，特别是根据极值定理，闭区间上的连续有界函数一定存在最大值和最小值。

10. Periodic Functions | 周期性

A function f is periodic with period T if f(x + T) = f(x) for all x in its domain, where T is a positive constant. The smallest positive such T is called the fundamental period.

若存在正常数 T，使得对所有定义域内的 x，都有 f(x + T) = f(x)，则称函数 f 是周期函数，T 称为周期。最小的正周期称为基本周期。

The trigonometric functions are the most common periodic functions:

三角函数是最常见的周期函数：

Function	Fundamental Period
sin x, cos x	2π
tan x	π
\|sin x\|	π

If a function has period T, then the function f(kx) has period T/|k|. For example, sin(2x) has period π, which is why the graph is compressed horizontally.

若函数 f 的周期为 T，则 f(kx) 的周期为 T/|k|。例如 sin(2x) 的周期为 π，所以图像在水平方向上被压缩了。

11. Transformations of Functions | 函数变换

Understanding how different transformations affect the graph of a function is essential for sketching and modelling. The table below summarises the most common transformations of y = f(x):

理解不同变换对函数图像的影响，对画图和建模至关重要。下表总结了 y = f(x) 最常见的几种变换：

Transformation	Equation	Effect on Graph
Vertical translation	y = f(x) + c	Shift up (c > 0) or down (c < 0)
Horizontal translation	y = f(x − a)	Shift right (a > 0) or left (a < 0)
Vertical stretch	y = k f(x)	Stretch vertically by factor k
Horizontal stretch	y = f(x/k)	Stretch horizontally by factor k
Reflection in x-axis	y = −f(x)	Flip over the x-axis
Reflection in y-axis	y = f(−x)	Flip over the y-axis

These transformations can be combined. For example, to sketch y = −2f(x − 1) + 3, first shift the graph of f one unit right, then stretch vertically by a factor of 2, then reflect in the x-axis, and finally shift up by 3. The order matters: transformations that affect x are applied in the opposite direction to what intuition may suggest.

这些变换可以组合使用。例如要画 y = −2f(x − 1) + 3，先把 f 的图像向右平移 1 个单位，然后纵向拉伸 2 倍，再关于 x 轴反射，最后向上平移 3 个单位。顺序很重要：影响 x 的变换与实际方向相反。

12. Summary and Exam Tips | 总结与考试提示

A function is a rule that assigns exactly one output to each input. Key concepts include domain, range, composition, inverse, graphs, symmetry, monotonicity, boundedness, periodicity, and transformations. Each property helps you understand the behaviour of a function more deeply.

函数是一种”每个输入恰好对应一个输出”的规则。核心概念包括定义域、值域、复合、反函数、图像、对称性、单调性、有界性、周期性和变换。每个性质都帮助你更深入地理解函数的行为。

For A-level exams, remember the following:

在 A-level 考试中，请记住以下几点：

Always state the domain and range when defining a function or its inverse.

在定义函数或反函数时，务必说明定义域和值域。
Check whether a function is one-to-one before finding its inverse.

求反函数之前，先确认函数是否为一一对应。
Use the vertical line test to determine whether a graph represents a function.

用铅垂线检验判断一个图像是否表示函数。
When composing functions, be careful about the domain of the composite expression.

求复合函数时，注意复合表达式本身的定义域。
Use derivatives to determine monotonicity and to locate critical points.

利用导数判断单调性并求极值点。
Know the graphs and periods of standard functions such as sin x, cos x, and tan x.

熟记 sin x、cos x、tan x 等标准函数的图像和周期。

Mastering these ideas will give you a solid foundation for calculus, series, and further mathematical study. Practice by sketching graphs, solving domain problems, and building composite and inverse functions until these methods become automatic.

掌握这些概念会为你学习微积分、级数和更深入的数学打下坚实基础。请通过画函数图像、求解定义域问题、构建复合函数与反函数来反复练习，直到这些方法运用自如。

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Functions: Concepts and Basic Properties | 函数的概念与基本性质