Inverse Functions: Definition and How to Find Them | 反函数的定义与求法

📚 Inverse Functions: Definition and How to Find Them | 反函数的定义与求法

In mathematics, a function maps each input to exactly one output. An inverse function reverses this process, sending each output back to the original input. Understanding inverse functions is essential in IB Mathematics, as it connects algebraic manipulation, graphical reasoning, and real-world applications.

在数学中，函数将每一个输入映射到唯一一个输出。反函数则逆转这一过程，将每一个输出送回原来的输入。理解反函数是IB数学的核心要求，它将代数运算、图像推理与实际应用紧密联系起来。

1. Revisiting the Definition of a Function | 回顾函数的定义

A function f from set A to set B is a rule that assigns each element x in A exactly one element y in B. We write y = f(x). The set A is called the domain, and the set of all outputs is called the range.

函数 f 是从集合 A 到集合 B 的一种规则，它将 A 中的每个元素 x 恰好分配给 B 中的一个元素 y。我们记作 y = f(x)。集合 A 称为定义域，所有输出值组成的集合称为值域。

For example, the function f(x) = 2x + 3 takes an input x, doubles it, and adds three. If x = 4, then f(4) = 11. The inverse should undo these operations in reverse order: subtract three and then divide by two.

例如，函数 f(x) = 2x + 3 将输入 x 加倍后加三。若 x = 4，则 f(4) = 11。反函数应当按相反顺序取消这些运算：先减三，再除以二。

2. The Intuitive Idea of an Inverse | 反函数的直观概念

An inverse function “undoes” what the original function does. If f takes x to y, then the inverse f⁻¹ takes y back to x. In other words, f⁻¹(y) = x when f(x) = y.

反函数“撤销”原函数所做的操作。如果 f 将 x 变为 y，那么反函数 f⁻¹ 将 y 变回 x。换句话说，当 f(x) = y 时，f⁻¹(y) = x。

Think of a machine that first adds 5 and then squares the result. To reverse it, you would first take the square root (if possible) and then subtract 5. This intuitive reversal is the basis of finding inverse functions algebraically.

可以把函数想象成一台机器：先加5，再平方。要逆向操作，你应当先开平方（如果可行），再减5。这种直观的逆向过程正是代数求反函数的基础。

3. Conditions for Existence: One-to-One Functions | 反函数存在的条件：一一对应函数

Not every function has an inverse. For an inverse to exist, each output must correspond to exactly one input. Such a function is called one-to-one (injective) or invertible.

并非所有函数都有反函数。要存在反函数，每一个输出值必须只对应一个输入值。这样的函数称为一一对应函数（单射）或可逆函数。

The horizontal line test is a graphical method to check this: if any horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one and does not have an inverse.

水平线检验是图形上的判断方法：如果任何一条水平线与函数图像相交多于一个点，则该函数不是一一对应的，也不存在反函数。

Important: For IB, you must state the domain restriction when a function is not naturally one-to-one, such as f(x) = x² restricted to x ≥ 0.

重要：在IB中，当函数本身不是一一对应时，你必须说明定义域限制，例如 f(x) = x² 限制在 x ≥ 0。

4. Formal Definition of an Inverse Function | 反函数的正式定义

Let f be a one-to-one function with domain A and range B. The inverse function f⁻¹ has domain B and range A, and is defined by the property:

设 f 是一个一一对应函数，定义域为 A，值域为 B。反函数 f⁻¹ 的定义域为 B，值域为 A，并满足以下性质：

f⁻¹(f(x)) = x for all x in A, and f(f⁻¹(y)) = y for all y in B.

This definition shows that composing a function with its inverse gives the identity function on the respective domain.

这一定义表明，函数与其反函数复合后，在各自定义域上得到恒等函数。

5. Step-by-Step Method to Find an Inverse | 求反函数的步骤

To find the inverse of a function given by y = f(x), follow these steps:

要求函数 y = f(x) 的反函数，请遵循以下步骤：

Step 1: Replace f(x) by y. / 第一步：将 f(x) 替换为 y。
Step 2: Swap x and y in the equation. / 第二步：在方程中交换 x 和 y。
Step 3: Solve the new equation for y in terms of x. / 第三步：解新方程，用 x 表示 y。
Step 4: Replace y by f⁻¹(x). / 第四步：将 y 替换为 f⁻¹(x)。
Step 5: State the domain of f⁻¹ as the range of f, and the range of f⁻¹ as the domain of f. / 第五步：说明 f⁻¹ 的定义域为 f 的值域，f⁻¹ 的值域为 f 的定义域。

This method works for any one-to-one function, including linear, quadratic (with domain restriction), exponential, logarithmic, and trigonometric functions (with restricted domains).

这一方法适用于任何一一对应函数，包括线性函数、二次函数（带定义域限制）、指数函数、对数函数以及三角函数（带限制定义域）。

6. Worked Example 1: A Linear Function | 实例1：线性函数

Consider f(x) = 2x – 5. Find f⁻¹(x).

设 f(x) = 2x – 5，求 f⁻¹(x)。

Step 1: Write y = 2x – 5. / 第一步：写 y = 2x – 5。

Step 2: Swap x and y: x = 2y – 5. / 第二步：交换 x 和 y：x = 2y – 5。

Step 3: Solve for y: x + 5 = 2y, so y = (x + 5)/2. / 第三步：解出 y：x + 5 = 2y，因此 y = (x + 5)/2。

f⁻¹(x) = (x + 5)/2

Check: f(f⁻¹(x)) = 2((x + 5)/2) – 5 = x + 5 – 5 = x. The inverse is correct.

验证：f(f⁻¹(x)) = 2((x + 5)/2) – 5 = x + 5 – 5 = x。反函数正确。

7. Worked Example 2: A Quadratic Function with a Restricted Domain | 实例2：限制定义域的二次函数

Let f(x) = x² + 1 with domain x ≥ 0. Find f⁻¹(x).

设 f(x) = x² + 1，定义域为 x ≥ 0。求 f⁻¹(x)。

First verify one-to-one: on the domain x ≥ 0, the graph is increasing, so each y-value comes from exactly one x-value. Now find the inverse.

首先验证一一对应：在定义域 x ≥ 0 上，图像递增，因此每个 y 值只对应一个 x 值。现在求反函数。

Set y = x² + 1. Swap x and y: x = y² + 1. Solve: y² = x – 1. Since the original domain is x ≥ 0, y must be non-negative, so we take the positive square root:

令 y = x² + 1。交换 x 和 y：x = y² + 1。解得 y² = x – 1。由于原定义域为 x ≥ 0，y 必须非负，因此取正平方根：

f⁻¹(x) = √(x – 1), with domain x ≥ 1.

Notice that the range of f is [1, ∞), which becomes the domain of f⁻¹, and the domain of f is [0, ∞), which becomes the range of f⁻¹.

注意 f 的值域为 [1, ∞)，它成为 f⁻¹ 的定义域；而 f 的定义域为 [0, ∞)，它成为 f⁻¹ 的值域。

8. Worked Example 3: Exponential and Logarithmic Functions | 实例3：指数函数与对数函数

The exponential function f(x) = 2ˣ is one-to-one on all real numbers. Its inverse is the logarithmic function with base 2.

指数函数 f(x) = 2ˣ 在整个实数范围内是一一对应的。它的反函数是以2为底的对数函数。

To find the inverse: y = 2ˣ. Swap: x = 2ʸ. Taking log₂ of both sides gives y = log₂x. Therefore:

求反函数：y = 2ˣ。交换：x = 2ʸ。两边取以2为底的对数得 y = log₂x。因此：

f⁻¹(x) = log₂x, for x > 0.

In general, for any base a > 0 and a ≠ 1, the inverse of f(x) = aˣ is f⁻¹(x) = logₐx. This relationship is fundamental in IB Mathematics.

一般来说，对任意底数 a > 0 且 a ≠ 1，f(x) = aˣ 的反函数是 f⁻¹(x) = logₐx。这一关系是IB数学中的基础。

9. Key Properties of Inverse Functions | 反函数的重要性质

Inverse functions have several important properties that are frequently tested in IB exams.

反函数有几个重要性质经常在IB考试中出现。

Domain and range swap: The domain of f⁻¹ equals the range of f, and the range of f⁻¹ equals the domain of f. / 定义域与值域互换：f⁻¹ 的定义域等于 f 的值域，f⁻¹ 的值域等于 f 的定义域。
Graph symmetry: The graph of f⁻¹ is the reflection of the graph of f across the line y = x. / 图像对称性：f⁻¹ 的图像是 f 的图像关于直线 y = x 的镜像。
Composition identity: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x, for all x in the appropriate domains. / 复合恒等：f(f⁻¹(x)) = x 且 f⁻¹(f(x)) = x，对所有在相应定义域内的 x 成立。
Inverse of an inverse: (f⁻¹)⁻¹ = f. / 反函数的反函数：(f⁻¹)⁻¹ = f。

For example, if f(x) = 2x + 3 and g(x) = (x – 3)/2, then f and g are inverses because f(g(x)) = 2((x – 3)/2) + 3 = x.

例如，若 f(x) = 2x + 3，g(x) = (x – 3)/2，则 f 和 g 互为反函数，因为 f(g(x)) = 2((x – 3)/2) + 3 = x。

10. Common Mistakes and IB Exam Tips | 常见错误与IB考点提示

Students often make errors when finding inverse functions. Here are the most common pitfalls.

学生在求反函数时常犯错误。以下是最常见的陷阱。

Forgetting to restrict the domain: For functions like x² or sin x, you must restrict the domain before an inverse can exist. / 忘记限制定义域：对于 x² 或 sin x 这样的函数，必须在求反函数前限制定义域。
Swapping variables incorrectly: Always swap x and y before solving; do not solve for y first and then swap. / 交换变量错误：务必先交换 x 和 y 再解方程；不要先解出 y 再交换。
Ignoring the domain of the inverse: The inverse’s domain is not automatically all real numbers; it is the range of the original function. / 忽略反函数的定义域：反函数的定义域不一定是所有实数，而是原函数的值域。
Confusing notation: f⁻¹(x) means the inverse function, not 1/f(x). / 混淆符号：f⁻¹(x) 表示反函数，不是 1/f(x)。

In IB exams, you should always check that your inverse satisfies f(f⁻¹(x)) = x, state the domain of f⁻¹ clearly, and draw the graph to confirm symmetry about y = x when time allows.

在IB考试中，你应当始终检查反函数满足 f(f⁻¹(x)) = x，明确说明 f⁻¹ 的定义域，并在时间允许时画图确认关于 y = x 的对称性。

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Inverse Functions: Definition and How to Find Them | 反函数的定义与求法