Functions: Domain, Range and Inverse | 函数：定义域、值域与反函数

📚 Functions: Domain, Range and Inverse | 函数：定义域、值域与反函数

Understanding functions is central to Edexcel A-Level Pure Mathematics. This unit brings together algebraic manipulation, graphical reasoning and domain/range analysis. Mastering it will help you tackle composite functions, inverse functions and transformations confidently.

理解函数是 Edexcel A-Level 纯数学的核心。本单元将代数运算、图形推理以及定义域和值域分析结合在一起。掌握它将帮助你自信地解决复合函数、反函数和函数变换问题。

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1. What is a Function? | 什么是函数？

A function is a mapping from a set of inputs, called the domain, to a set of outputs, called the codomain, such that each input is assigned exactly one output. In Edexcel papers, functions are usually defined by a formula and a specified domain.

函数是从输入集合（称为定义域）到输出集合（称为陪域）的映射，使得每个输入恰好对应一个输出。在 Edexcel 试卷中，函数通常由公式和指定的定义域来定义。

The notation f(x) = x² + 3 means take x, square it, then add 3. A function can also be written as f : x ↦ x² + 3.

符号 f(x) = x² + 3 表示取 x，平方后再加 3。函数也可以写作 f : x ↦ x² + 3。

f(x) = x² + 3, x ∈ ℝ

Two functions are the same only if they have the same rule and the same domain. Changing the domain creates a different function even if the formula is unchanged.

两个函数只有拥有相同的规则和相同的定义域才是同一个函数。即使公式不变，改变定义域也会产生不同的函数。

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2. Domain and Range | 定义域与值域

The domain is the set of all possible input values for which the function is defined. The range is the set of all outputs the function actually produces.

定义域是使函数有定义的所有可能输入值的集合。值域是函数实际产生的所有输出的集合。

For f(x) = x², x ∈ ℝ, the domain is all real numbers, but the range is f(x) ≥ 0 because squares cannot be negative.

对于 f(x) = x²，x ∈ ℝ，定义域是所有实数，但值域是 f(x) ≥ 0，因为平方数不可能为负。

Domain = ℝ, Range = [0, ∞)

When a domain is restricted, the range must be recalculated using the endpoints and any turning points within the interval. For example, f(x) = x² – 4x + 5 on x ∈ [0, 3] has a minimum at x = 2, giving the range [1, 5].

当定义域受到限制时，必须使用区间端点和区间内的任何驻点重新计算值域。例如，f(x) = x² – 4x + 5 在 x ∈ [0, 3] 上的最小值在 x = 2 处，因此值域为 [1, 5]。

You must use correct notation: x ∈ ℝ, x ≥ 0 for domain; f(x) ∈ ℝ, f(x) > 0 for range. Edexcel accepts inequalities or interval notation.

必须使用正确的符号：定义域写作 x ∈ ℝ，x ≥ 0；值域写作 f(x) ∈ ℝ，f(x) > 0。Edexcel 接受不等式或区间表示法。

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3. Composite Functions | 复合函数

A composite function applies one function after another. fg(x) means f(g(x)), so you first apply g to x, then apply f to the result.

复合函数是先应用一个函数，再应用另一个函数。fg(x) 表示 f(g(x))，因此你先对 x 应用 g，然后对结果应用 f。

The order matters because fg(x) and gf(x) are generally different. Always follow the order indicated by the notation.

顺序很重要，因为 fg(x) 和 gf(x) 通常是不同的。务必遵循符号所示的顺序。

For fg(x) to exist at a particular x, the output g(x) must lie in the domain of f. The domain of fg is therefore x ∈ domain of g such that g(x) ∈ domain of f.

要使 fg(x) 在某个 x 处存在，输出 g(x) 必须落在 f 的定义域内。因此 fg 的定义域是 x ∈ g 的定义域，并且 g(x) ∈ f 的定义域。

Example: f(x) = 2x + 1 and g(x) = √x give fg(x) = f(√x) = 2√x + 1 with domain x ≥ 0. Also gf(x) = g(2x + 1) = √(2x + 1) with domain x ≥ -½.

示例：f(x) = 2x + 1 和 g(x) = √x 得到 fg(x) = f(√x) = 2√x + 1，定义域为 x ≥ 0。同时 gf(x) = g(2x + 1) = √(2x + 1)，定义域为 x ≥ -½。

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4. Inverse Functions | 反函数

The inverse function f⁻¹(x) reverses the effect of f. If f(a) = b, then f⁻¹(b) = a. A function only has an inverse if it is one-to-one on its domain.

反函数 f⁻¹(x) 逆转 f 的作用。如果 f(a) = b，那么 f⁻¹(b) = a。只有函数在其定义域上是一一对应的，它才存在反函数。

To find an inverse, write y = f(x), rearrange to make x the subject, then swap x and y. The result is y = f⁻¹(x).

要求反函数，先写出 y = f(x)，整理出 x，然后交换 x 和 y。所得结果就是 y = f⁻¹(x)。

The domain of f⁻¹ is the range of f, and the range of f⁻¹

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