📚 The Constitution of Functions: Domain, Range, Composite & Inverse | 函数的构成：定义域、值域、复合与反函数
In Edexcel A-Level Mathematics, studying functions goes beyond memorising formulas. It requires understanding how a function is constituted: its domain, its range, the mapping rule, and the way functions can be combined or reversed. This article breaks down the constitution of functions and builds the key skills needed for success in Pure Mathematics.
在 Edexcel A-Level 数学中，学习函数不只是记忆公式。它要求理解函数是如何构成的：定义域、值域、对应法则，以及函数如何被复合或逆转。本文解析函数的构成，并建立纯数学考试成功所需的关键技能。
1. What is the Constitution of a Function? | 函数的构成是什么？
A function is a special mapping from a set of inputs, called the domain, to a set of outputs, called the range, where each input is assigned exactly one output. The constitution of a function therefore has three essential parts: the domain, the range, and the rule linking them.
函数是从输入集合（称为定义域）到输出集合（称为值域）的一种特殊映射，其中每个输入恰好对应一个输出。因此，函数的构成包含三个基本部分：定义域、值域以及连接它们的法则。
For example, the rule f(x) = 2x + 1 can be written in mapping notation as follows:
例如，法则 f(x) = 2x + 1 可以用映射记号写成：
f: x → 2x + 1, x ∈ ℝ
This notation tells us that any real number x is mapped to the value 2x + 1. The domain is the set of all real numbers, written as ℝ, and the range is also all real numbers because 2x + 1 can take any real value.
这个记号告诉我们，任何实数 x 都被映射到值 2x + 1。定义域是全体实数，记作 ℝ；值域也是全体实数，因为 2x + 1 可以取到任意实数值。
2. Domain and Range: The Core Components | 定义域与值域：核心要素
e set of all allowed input values. For real-valued functions, this is often expressed as x ∈ ℝ mposes restrictions. For example, f(x) = 1/(x − 2) cannot accept x = 2, so its domain is x ∈ ℝ, x ≠ 2.
定义域是所有允许的输入值的集合。对于实值函数，通常表示为 x ∈ ℝ，除非分母、平方根或对数施加了限制。例如，f(x) = 1/(x − 2) 不能取 x = 2，因此其定义域为 x ∈ ℝ 且 x ≠ 2。
lete the square. For f(x) = x², the domain is all real numbers, but the range is f(x) ≥ 0.
值域是应用函数法则后所有可能输出值的集合。求值域通常需要结合图像形状或配方法。对于 f(x) = x²，定义域为全体实数，但值域为 f(x) ≥ 0。
When a question asks for the domain, check for three common restrictions: denominators cannot be zero, expressions under an even root must be non-negative, and logarithms require positive inputs.
当题目要求求定义域时，检查三种常见限制：分母不能为零；偶次根号下的表达式必须非负；对数函数要求输入为正。
For the range, consider whether the function can produce all real numbers or only a subset. Quadratic functions, for instance, have a maximum or minimum output that limits the range.
对于值域，要考虑函数能否产生所有实数，还是只能产生某个子集。例如，二次函数有最大值或最小值，这会限制值域。
3. Function Notation and Mapping Diagrams | 函数记号与映射图
Function notation f(x) is used to show the output of a function named f at a given input x. The letter f is the name of the function, while f(x) is the value it assigns to x. This notation allows us to evaluate the function at specific points and to compose functions later.
函数记号 f(x) 用于表示名为 f 的函数在给定输入 x 时的输出。字母 f 是函数的名称，而 f(x) 是它赋予 x 的值。这种记号使我们能够在特定点求值，并在之后进行函数复合。
, and so on. No input can map to two different outputs, but two inputs may share the same output.
映射图将每个输入元素与其唯一输出元素的对应关系可视化。例如，若 f(x) = x + 2，则 x = 1 映射到 3，x = 2 映射到 4，依此类推。任何一个输入不能对应两个不同输出，但两个输入可以共享同一个输出。
A useful way to think of a function is as a machine: you put a number in, the machine applies the rule, and it produces exactly one output. This picture helps when forming composite functions, where the output of one machine becomes the input of another.
理解函数的一个有效方式是将其想象成一台机器：放入一个数字，机器应用规则，并准确产生一个输出。这个图像在构造复合函数时非常有用，因为一台机器的输出会成为另一台机器的输入。
In exam questions, you will often see notation such as f: x → x² − 3x, x ∈ ℝ into x² − 3x. Always read the domain stated alongside the rule; it is part of the function's constitution.
在考试题中，你经常会看到诸如 f: x → x² − 3x, x ∈ ℝ 的记号。这表示函数 f 取一个实数 x 并将其变换为 x² − 3x。始终要阅读法则旁边给出的定义域；它是函数构成的一部分。
4. Types of Functions: One-to-One and Many-to-One | 函数类型：一一映射与多对一映射
s one-to-one. Linear functions such as f(x) = 3x − 5 are one-to-one over all real numbers.
一一映射函数中每个输出都有唯一的输入。代数上，如果 f(a) = f(b) 推出 a = b，那么 f 就是一一映射。线性函数如 f(x) = 3x − 5 在全体实数上都是一一映射。
4. Many quadratic and trigonometric functions are many-to-one over their natural domains.
多对一函数允许多个不同输入产生相同输出，例如 f(x) = x²，其中 x = 2 和 x = −2 都给出 f(x) = 4。许多二次函数和三角函数在其自然定义域上是多对一的。
Only one-to-one functions have inverse functions over their whole domain. A many-to-one function can still have an inverse if its domain is restricted so that it becomes one-to-one.
只有一一映射函数才能在整个定义域上具有反函数。多对一函数可以通过限制定义域使其变成一一映射，从而具有反函数。
A horizontal line test on the graph is a quick way to check the type: if any horizontal line cuts the graph more than once, the function is many-to-one. If every horizontal line cuts the graph at most once, it is one-to-one.
对图像作水平线检验是判断类型的一种快速方法：若任何水平线与图像的交点多于一个，则函数是多对一的；若每条水平线最多与图像相交一次，则函数是一一映射。
5. Composite Functions: Building New Functions | 复合函数：构造新函数
tation fg(x) means apply g first, then apply f to the result. In symbols, fg(x) = f(g(x)).
当一个函数的输出成为另一个函数的输入时，就形成了复合函数。记号 fg(x) 表示先应用 g，再将 f 作用于结果。用符号表示就是 fg(x) = f(g(x))。
xample, if f(x) = x + 2 and g(x) = 3x, then fg(x) = 3x + 2, but gf(x) = 3(x + 2) = 3x + 6.
复合的顺序很重要。对大多数函数而言，gf(x) 与 fg(x) 是不同的。例如，若 f(x) = x + 2，g(x) = 3x，则 fg(x) = 3x + 2，而 gf(x) = 3(x + 2) = 3x + 6。
ou can also form composite functions with three or more functions. For instance, if h(x) = √x, then hgf(x) means apply f first, then g, then h. In practice, always work from the innermost function outward.
你还可以用三个或更多函数构造复合函数。例如，若 h(x) = √x，则 hgf(x) 表示先应用 f，再应用 g，最后应用 h。实际操作中，总是从最内层的函数开始向外计算。
When writing composite functions, be careful with parentheses. For fg(x), the input x goes into g first, so g(x) must be well-defined before f is applied. Never assume that fg(x) equals gf(x) without checking.
书写复合函数时要注意括号。对于 fg(x)，输入 x 首先进入 g，因此在应用 f 之前必须确保 g(x) 有定义。未经验证不要假设 fg(x) 等于 gf(x)。
6. Domain Rules for Composite Functions | 复合函数的定义域规则
For fg(x) to exist
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