📚 Exercise 2G: Inverse Functions | 练习2G：反函数
Welcome to the IB Mathematics study guide for Exercise 2G, which focuses on inverse functions. In your textbook, Exercise 2G is designed to build your skills in reversing function operations, stating domains and ranges, and sketching the graphical relationship between a function and its inverse. This revision article unpacks every essential concept you need to master inverse functions.
欢迎来到IB数学关于练习2G的学习指南，主题是反函数。在你的教材中，练习2G旨在培养你逆转函数运算、写出定义域与值域以及绘制函数与其反函数之间图形关系的能力。这篇复习文章将分解你掌握反函数所需的每一个基本概念。
1. What is an Inverse Function? | 什么是反函数？
⁻¹ to be a function, each y-value must be paired with exactly one x-value, which means the original function must be one-to-one (injective).
反函数逆转原函数的作用。如果函数 f 将输入 x 映射到输出 y (y = f(x))，那么它的反函数记作 f⁻¹，将 y 送回到 x，即 x = f⁻¹(y)。要使得 f⁻¹ 成为一个函数，每个 y 值必须恰好对应一个 x 值，这意味着原函数必须是一一对应的（单射）。
The idea of 'undoing' (f(x)) = x for all x in the domain of f, and f(f⁻¹(x)) = x for all x in the domain of f⁻¹.
'撤销'的概念可以用复合恒等式表达：对 f 定义域内的所有 x，f⁻¹(f(x)) = x；对 f⁻¹ 定义域内的所有 x，f(f⁻¹(x)) = x。
2. The Horizontal Line Test | 水平线检验
To decide whether a function's inverse is itself a function without restricting the domain, we apply the horizontal line test. If any horizontal line cuts the graph of f more than once, then f is not one-to-one, and its inverse will not pass the vertical line test, meaning it is not a function.
要判断一个函数的反函数本身是否是一个函数（不限制定义域），我们使用水平线检验。如果任意一条水平线与 f 的图像相交超过一次，那么 f 就不是一一对应的，其反函数将无法通过垂直线检验，也就是它不是一个函数。
ce, the parabola f(x) = x² fails the horizontal line test, so the square root relation y = √x is only a function if we restrict the domain of f(x) = x² to x ≥ 0.
例如，抛物线 f(x) = x² 未能通过水平线检验，因此平方根关系 y = √x 仅当我们将 f(x) = x² 的定义域限制为 x ≥ 0 时才是一个函数。
3. One-to-One Functions | 一一对应函数
es both the vertical and the horizontal line tests. Only one-to-one functions have inverses that are also functions without needing extra restrictions. In IB problems, you will often be asked to restrict the domain of a non-one-to-one function to make it invertible, then find and describe its inverse.
如果 f(a) = f(b) 蕴含 a = b，则函数 f 是一一对应的。从图像上看，它同时通过垂直线检验和水平线检验。只有一一对应函数的反函数是不需要额外限制的函数。在IB题目中，你常常会被要求限制一个非一一对应函数的定义域以使其可逆，然后求出并描述其反函数。
4. Notation: f⁻¹(x) | 符号：f⁻¹(x)
mbol for the inverse function, not an exponent indicating a reciprocal. Always read f⁻¹ as 'f inverse' if f(x) = 2x + 5, then f⁻¹(x) = (x - 5)/2, which is completely different from 1/(2x + 5).
符号 f⁻¹(x) 常被误解；它并不表示 1/f(x)。上标 -1 仅仅是反函数的符号，而不是表示倒数的指数。请始终将 f⁻¹ 读作 'f 逆'。例如，若 f(x) = 2x + 5，则 f⁻¹(x) = (x - 5)/2，这与 1/(2x + 5) 完全不同。
5. Domain and Range Swap | 定义域与值域的互换
A vital property of inverse functions is that the domain of f becomes the range of f⁻¹, and the range of f becomes the domain of f⁻¹. If f has domain D and range R, then f⁻¹ has domain R and range D. This symmetrical relationship is extremely useful when determining possible input and output values without explicitly solving for the inverse.
反函数的一个关键性质是：f 的定义域变成 f⁻¹ 的值域，而 f 的值域变成 f⁻¹ 的定义域。如果 f 的定义域为 D，值域为 R，那么 f⁻¹ 的定义域就是 R，值域就是 D。这种对称关系在不用显式求出反函数的情况下确定可能的输入和输出值时非常有用。
You can use this swap to check your algebraic working: after finding f⁻¹, verify that its domain matches the range of the original function.
你可以利用这种互换来检验代数运算：求出 f⁻¹ 后，验证其定义域是否与原函数的值域一致。
6. Finding the Inverse Function Algebraically | 代数法求反函数
The standard algebraic method for finding an inverse function consists of three clear steps.
求反函数的标准代数方法由三个明确的步骤组成。
- Step 1: Write the function as y = f(x).
- 第1步：将函数写成 y = f(x)。
- Step 2: Swap x and y, producing the equation x = f(y).
- 第2步：交换 x 和 y，得到方程 x = f(y)。
- Step 3: Solve this new equation for y. The resulting expression is y = f⁻¹(x).
- 第3步：解这个新方程求出 y。得到的表达式就是 y = f⁻¹(x)。
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图