Inverse Functions: Domain, Graphs and Exam Skills | 反函数：定义域、图像与应试技巧

📚 Inverse Functions: Domain, Graphs and Exam Skills | 反函数：定义域、图像与应试技巧

In Edexcel A-Level Mathematics, inverse functions appear throughout Pure 1, Pure 2 and Pure 3. They are used to reverse mappings, solve equations, and understand symmetry between functions such as exponentials and logarithms. This article covers the definitions, algebraic methods, graphs, common exam traps and worked examples you need to master.

在 Edexcel A-Level 数学中，反函数贯穿 Pure 1、Pure 2 和 Pure 3。它们用于逆转映射、解方程，以及理解指数与对数等函数之间的对称关系。本文涵盖定义、代数方法、图像、常见考试陷阱和你必须掌握的例题。

1. What Is an Inverse Function? | 什么是反函数？

A function f maps an input x to an output y = f(x). If this mapping can be reversed, the reverse rule is called the inverse function, written f⁻¹. It satisfies f⁻¹(f(x)) = x for every x in the domain of f, and f(f⁻¹(x)) = x for every x in the domain of f⁻¹.

函数 f 将输入 x 映射为输出 y = f(x)。如果这种映射可以逆转，反转后的规则就称为反函数，记作 f⁻¹。它满足 f⁻¹(f(x)) = x（对 f 定义域内每个 x 成立），以及 f(f⁻¹(x)) = x（对 f⁻¹ 定义域内每个 x 成立）。

In simple terms, f⁻¹ undoes what f does. If f turns 3 into 7, then f⁻¹ turns 7 back into 3.

简单地说，f⁻¹ 会撤销 f 的作用。如果 f 把 3 变成 7，那么 f⁻¹ 就把 7 变回 3。

2. One-to-One Functions and the Horizontal Line Test | 一一函数与水平线检验

A function has an inverse only if it is one-to-one, also called injective. This means no two different inputs produce the same output. Graphically, any horizontal line y = k should intersect the graph y = f(x) at most once. This is the horizontal line test.

函数只有在一一对应（单射）时才存在反函数。这意味着不存在两个不同输入产生相同输出。从图像上看，任意水平线 y = k 与曲线 y = f(x) 至多相交一次，这就是水平线检验。

For example, f(x) = x² with domain all real numbers is not one-to-one because f(2) = f(-2) = 4. However, if the domain is restricted to x ≥ 0, then f becomes one-to-one and an inverse exists.

例如，f(x) = x² 定义在全体实数上不是一一函数，因为 f(2) = f(-2) = 4。但如果把定义域限制为 x ≥ 0，则 f 变为一一函数，反函数存在。

3. Domain and Range Swap | 定义域与值域的交换

The domain of f becomes the range of f⁻¹, and the range of f becomes the domain of f⁻¹. This swap is fundamental when stating an inverse function fully.

f 的定义域成为 f⁻¹ 的值域，f 的值域成为 f⁻¹ 的定义域。在完整写出反函数时，这种交换是核心。

For f(x) = 2x + 3 with domain x ∈ ℝ, the range is also ℝ. Therefore f⁻¹(x) = (x – 3)/2 has domain ℝ and range ℝ

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