Inverse Functions: Concepts and Methods | 反函数的概念与求法

📚 Inverse Functions: Concepts and Methods | 反函数的概念与求法

In mathematics, the idea of an inverse appears everywhere: addition and subtraction undo each other, just as multiplication and division do. For functions, the inverse undoes the original mapping and recovers the starting input. Mastering inverse functions is essential for IB Mathematics, since the concept reappears in algebra, calculus, and many real-world applications.

在数学中，逆运算的概念无处不在：加法与减法互为逆运算，乘法与除法也互为逆运算。对于函数而言，反函数的作用就是撤销原来的映射，找回最初的输入值。掌握反函数是 IB 数学学习的核心要求，因为这一概念在代数、微积分以及许多实际应用中反复出现。

1. Functions and Mappings | 函数与映射

A function f assigns to each input exactly one output. We denote a function by f: A → B or y = f(x), where A is the domain and B is the codomain.

函数 f 把每一个输入值对应到唯一一个输出值。我们用 f: A → B 或 y = f(x) 来表示函数，其中 A 是定义域，B 是陪域。

The domain is the set of allowed inputs; the range is the subset of outputs that f actually produces from those inputs.

定义域是允许的输入值集合；值域是 f 根据这些输入值实际产生的输出值所构成的集合。

For an inverse to exist, the original function must be one-to-one; otherwise the “inverse” would not satisfy the definition of a function.

要使反函数存在，原函数必须是一一对应的；否则这个所谓的“反函数”将不满足函数的定义。

2. Definition of Inverse Function | 反函数的定义

If f: A → B is a function, its inverse f⁻¹: B → A satisfies f⁻¹(f(x)) = x for every x in A, and f(f⁻¹(y)) = y for every y in B.

若 f: A → B 是一个函数，它的反函数 f⁻¹: B → A 满足：对 A 中任意 x，有 f⁻¹(f(x)) = x；对 B 中任意 y，有 f(f⁻¹(y)) = y。

In simpler terms, if f sends a to b, then f⁻¹ sends b back to a. The inverse undoes the original operation.

简单地说，如果 f 把 a 变为 b，那么 f⁻¹ 就把 b 还原为 a。反函数本质上撤销了原来的运算。

Do not confuse f⁻¹(x) with the reciprocal 1/f(x). The raised −1 denotes the inverse function, not an exponent.

千万不要把 f⁻¹(x) 与倒数 1/f(x) 混淆。这里的上标 −1 表示反函数，而不是指数。

3. One-to-One Functions | 一一对应的函数

A function is one-to-one, or injective, if f(a) = f

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