Functions and Mappings | 函数与映射

📚 Functions and Mappings | 函数与映射

Functions and mappings are fundamental to Edexcel A-Level Pure Mathematics. This revision guide explains the key definitions, notation, graphical tests and algebraic techniques you need for domain, range, composite functions and inverse functions.

函数与映射是 Edexcel A-Level 纯数学的基础内容。本复习指南讲解关键定义、符号、图像检验以及处理定义域、值域、复合函数和反函数所需的代数技巧。

1. Mapping Terminology | 映射术语

A mapping takes each element of a starting set and assigns it to an element of a finishing set. In A-Level questions, the starting set is usually called the domain and the finishing set is the codomain.

映射将起始集合中的每个元素对应到终止集合中的一个元素。在 A-Level 考题中，起始集合通常称为定义域，终止集合称为上域。

The range is the set of outputs that the mapping actually produces from the given domain. It is always a subset of the codomain, so the two terms must not be confused.

值域是映射在给定定义域下实际产生的所有输出组成的集合。它总是上域的子集，因此这两个术语不能混淆。

Domain → Mapping → Range ⊆ Codomain

2. What Makes a Function? | 什么构成了函数？

A function is a mapping with one crucial rule: every input in the domain must have exactly one output. If a single input can produce two different outputs, the mapping is not a function.

函数是一种映射，但有一条核心规则：定义域中的每个输入必须恰好有一个输出。如果一个输入可能产生两个不同输出，那么该映射就不是函数。

Therefore, one-to-one and many-to-one mappings can be functions, but one-to-many mappings cannot. Graphically, a curve represents a function only if any vertical line crosses it at most once.

因此，一对一和多对一映射可以构成函数，而一对多映射不能。从图像上看，只有当任意一条竖直线与曲线至多相交一次时，该曲线才表示一个函数。

3. Domain, Codomain and Range | 定义域、上域和值域

The domain is the set of allowed input values, often written as x ∈ ℝ, x > 0, x ≠ 2 or using interval notation. The codomain is stated by the function definition; the range must be calculated or deduced from the graph.

定义域是允许输入的集合，通常写作 x ∈ ℝ、x > 0、x ≠ 2 或使用区间表示。上域由函数定义给出，而值域需要通过计算或图像推断。

For example, f(x) = x² with domain x ∈ ℝ has codomain ℝ, but its range is [0, ∞) because squares are never negative. Restrictions such as square roots, logarithms and denominators often limit the domain.

例如，f(x) = x² 的定义域为 x ∈ ℝ，上域为 ℝ，但其值域为 [0, ∞)，因为平方永远不会为负。平方根、对数和分母等限制通常会缩小定义域。

4. Function Notation and Evaluation | 函数符号与求值

Function notation f(x) = 3x − 5 means that the rule is applied to an input x. To evaluate f(2), replace x by 2: f(2) = 3(2) − 5 = 1.

函数符号 f(x) = 3x − 5 表示把该规则作用于输入 x。求 f(2) 时，用 2 替换 x：f(2) = 3(2) − 5 = 1。

You may also see mapping notation f : x ↦ 3x − 5. Solving f(x) = k means setting the rule equal to k and finding the input values that satisfy it.

你还会看到映射符号 f : x ↦ 3x − 5。解方程 f(x) = k 就是令规则等于 k，并找出满足该方程的输入值。

5. One-to-One and Many-to-One Functions | 一对一与多对一函数

A one-to-one function, also called injective, gives a different output for every different input. Its graph passes the horizontal line test: any horizontal line meets the graph at most once.

一对一函数又称单射函数，每个不同输入都对应不同的输出。其图像通过水平线检验：任何水平线与图像至多相交一次。

A many-to-one function allows two or more inputs to give the same output, such as f(x) = x². These functions are still valid, but they do not have inverses unless the domain is restricted first.

多对一函数允许多个输入产生相同输出，例如 f(x) = x²。这类函数仍然是有效函数，但除非先限制定义域，否则它们没有反函数。

6. Composite Functions | 复合函数

The composite function gf(x) means apply f first, then apply g to the result. Algebraically, gf(x) = g(f(x)). The order matters: fg(x) is usually different from gf(x).

复合函数 gf(x) 表示先作用 f，再将结果代入 g。代数上，gf(x) = g(f(x))。顺序很重要：fg(x) 通常与 gf(x) 不同。

The domain of gf consists of inputs x that belong to the domain of f and for which f(x) lies in the domain of g. This is a common Edexcel pitfall.

复合函数 gf 的定义域由既属于 f 的定义域、又能使 f(x) 落在 g 的定义域中的输入组成。这是 Edexcel 考试中的常见失分点。

gf(x) = g(f(x)) ≠ f(g(x))

7. Inverse Functions | 反函数

The inverse function f⁻¹(x) reverses the effect of f. Only one-to-one functions have an inverse over their whole domain; a many-to-one function must first have its domain restricted.

反函数 f⁻¹(x) 逆转函数 f 的作用。只有一对一函数在整个定义域上才有反函数；多对一函数必须先限制定义域。

To find an inverse, write y = f(x), rearrange to make x the subject, then swap x and y. The domain and range of f and f⁻¹ are swapped.

求反函数时，先写 y = f(x)，整理使 x 成为主项，然后交换 x 和 y。f 与 f⁻¹ 的定义域和值域互相交换。

f(x) = 2x + 3 → f⁻¹(x) = (x − 3)/2

8. Restricting the Domain | 限制定义域

To make a many-to-one function invertible, choose a section of the graph that passes the horizontal line test. For f(x) = x², the usual restriction is x ≥ 0, giving f⁻¹(x) = √x.

要使多对一函数可逆，需选取图像中通过水平线检验的一段。对于 f(x) = x²，通常限制为 x ≥ 0，从而得到 f⁻¹(x) = √x。

For quadratic functions, complete the square to find the vertex. Restricting to one side of the vertex, such as x ≥ 2 for f(x) = (x − 2)², creates a one-to-one function.

对于二次函数，可通过配平方找到顶点。限制在顶点的一侧，例如 f(x) = (x − 2)² 取 x ≥ 2，就可构造出一对一函数。

9. Modulus Functions as Mappings | 绝对值函数作为映射

The modulus function f(x) = |x| is defined by |x| = x when x ≥ 0 and |x| = −x when x < 0. Its graph is a V-shape with vertex at the origin and range y ≥ 0.

绝对值函数 f(x) = |x| 定义为：当 x ≥ 0 时 |x| = x，当 x < 0 时 |x| = −x。其图像是以原点为顶点的 V 形，值域为 y ≥ 0。

Because |−3| = |3| = 3, the modulus mapping is many-to-one. It can be used inside composite functions, and solving modulus equations often produces two cases.

由于 |−3| = |3| = 3，绝对值映射是多对一的。它可出现在复合函数中，解绝对值方程时通常会产生两种情形。

10. Graphical Tests and Exam Tips | 图像检验与考试技巧

Use the vertical line test to confirm a graph represents a function, and the horizontal line test to check whether it is one-to-one. Sketching graphs helps identify the range and any restrictions.

使用竖直线检验确认图像是否表示函数，使用水平线检验判断是否一对一。画草图有助于确定值域和任何限制。

Common mistakes include confusing range with codomain, forgetting to check the domain of a composite function, and finding an inverse without first proving the function is one-to-one. Always write domain and range using correct set notation.

常见错误包括混淆值域与上域、忘记检查复合函数的定义域，以及在未证明函数一对一之前就求反函数。务必使用正确的集合符号书写定义域和值域。

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Functions and Mappings | 函数与映射