Inverse Functions | 反函数

📚 Inverse Functions | 反函数

In inverse functions are a core topic in Edexcel IGCSE Mathematics. When you understand how to ‘reverse’ a function, you unlock a powerful tool for solving equations, analysing graphs, and working with composite functions. In this revision guide, we will explore the definition, notation, step-by-step techniques, graphical properties, and exam strategies for inverse functions.

反函数是 Edexcel IGCSE 数学中的核心考点。当你理解如何“反过来”操作一个函数时，你就掌握了一个强大的工具，可用于解方程、分析图像以及处理复合函数。在本复习指南中，我们将深入探讨反函数的定义、符号、求解步骤、图像性质与考试策略。

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

A function maps an input x to an output f(x). An inverse function reverses this mapping: it takes the output and returns the original input. In other words, if f(a) = b, then f⁻¹(b) = a.

函数将输入 x 映射到输出 f(x)。反函数则逆转这一映射：它接受输出并返回原来的输入。换句话说，如果 f(a) = b，那么 f⁻¹(b) = a。

f⁻¹(f(x)) = x and f(f⁻¹(x)) = x

This property is the fundamental test for an inverse function. No matter what value of x you choose, applying f and then f⁻¹ (or applying f⁻¹ and then f) returns you to the original value.

这个性质是验证反函数的基本判据。无论选择什么 x 值，先应用 f 再应用 f⁻¹（或先 f⁻¹ 再 f）都会回到原始值。

2. Notation: f⁻¹(x) | 符号：f⁻¹(x)

The inverse of f(x) is written as f⁻¹(x). The superscript -1 is not an exponent; it means ‘inverse’. This is a common source of confusion in exams.

函数 f(x) 的反函数记作 f⁻¹(x)。上标 -1 不是指数，而是表示“反函数”。这是考试中常见的混淆点。

f⁻¹(x) ≠ 1/f(x)

For example, if f(x) = 3x, then f⁻¹(x) = x/3, not 1/(3x). Always read the notation carefully before answering.

例如，若 f(x) = 3x，那么 f⁻¹(x) = x/3，而不是 1/(3x)。答题前一定要仔细分辨这个符号。

3. Step-by-Step Method to Find the Inverse | 求反函数的步骤

To find the inverse of a function f(x), follow these simple steps.

求函数 f(x) 的反函数，请遵循以下简单步骤。

Write y = f(x).

写出 y = f(x)。
Swap x and y in the equation.

将方程中的 x 与 y 互换。
Rearrange the equation to make y the subject.

重新整理方程，使 y 成为主项。
Replace y with f⁻¹(x).

将 y 替换为 f⁻¹(x)。

Why do we swap x and y? Because the inverse function needs to send outputs back to inputs. Swapping the variables mathematically encodes this reversal.

为什么要交换 x 和 y？因为反函数需要把输出送回原来的输入。交换变量就是在数学上实现这种逆转。

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

Find the inverse of f(x) = 2x + 3.

求 f(x) = 2x + 3 的反函数。

Step 1: Write y = 2x + 3.

第一步：写出 y = 2x + 3。
Step 2: Swap x and y: x = 2y + 3.

第二步：交换 x 和 y：x = 2y + 3。
Step 3: Rearrange: 2y = x – 3, so y = (x – 3)/2.

第三步：整理得到 2y = x – 3，所以 y = (x – 3)/2。

f⁻¹(x) = (x – 3)/2

You can check this result by composing: f⁻¹(f(x)) = f⁻¹(2x + 3) = ((2x + 3) – 3)/2 = x. The inverse works.

你可以通过复合来验证：f⁻¹(f(x)) = f⁻¹(2x + 3) = ((2x + 3) – 3)/2 = x。这个反函数是正确的。

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

Not every function has an inverse without a restricted domain. Consider f(x) = x² with domain x ≥ 0.

并非每个函数不加限制就有反函数。考虑定义域为 x ≥ 0 的 f(x) = x²。

Step 1: Write y = x².

第一步：写出 y = x²。
Step 2: Swap x and y: x = y².

第二步：交换 x 和 y：x = y²。
Step 3: Solve for y. Since the original domain is x ≥ 0, the output y ≥ 0, so we choose the positive root: y = √x.

第三步：解出 y。因为原函数定义域 x ≥ 0，输出 y ≥ 0，所以我们取正根：y = √x。

f⁻¹(x) = √x, x ≥ 0

Without the domain restriction x ≥ 0, the function y = x² would not be one-to-one and would fail the horizontal line test.

如果没有定义域限制 x ≥ 0，函数 y = x² 就不是一对一的，也无法通过水平线检验。

6. Domain and Range Swap | 定义域与值域互换

One of the most important relationships between a function and its inverse is that the domain and range swap. If f maps set A to set B, then f⁻¹ maps set B back to set A.

函数与其反函数之间最重要的关系之一是定义域与值域互换。若 f 将集合 A 映射到集合 B，则 f⁻¹ 将集合 B 映射回集合 A。

Function	Domain	Range
f(x) = √x	x ≥ 0	y ≥ 0
f⁻¹(x) = x²	x ≥ 0	y ≥ 0

In this example, the domain and range happen to be the same interval, but for many functions, such as exponential and logarithmic pairs, the swap is easy to see.

在这个例子中，定义域与值域恰好是同一个区间；但对于许多函数，比如指数函数与对数函数这一对，互换关系会更明显。

7. Graphical Interpretation: Reflection in y = x | 图像解释：关于 y = x 对称

The graph of an inverse function is the reflection of the graph of the original function in the line y = x. If the point (a, b) lies on the graph of f, then the point (b, a) lies on the graph of f⁻¹.

反函数的图像是原函数图像关于直线 y = x 的对称图形。如果点 (a, b) 在 f 的图像上，那么点 (b, a) 就在 f⁻¹ 的图像上。

(a, b) on f ⇔ (b, a) on f⁻¹

This means that the x-coordinate and y-coordinate of every point are swapped. When drawing an inverse graph, first plot the line y = x, then reflect a few key points to get an accurate curve.

这意味着每个点的横坐标与纵坐标都会被交换。在画反函数图像时，先画出直线 y = x，然后取几个关键点进行对称，就能得到准确的曲线。

8. The Horizontal Line Test | 水平线检验

To decide whether a function has an inverse, use the horizontal line test. A function has an inverse if and only if every horizontal line intersects its graph at most once. Such functions are called one-to-one or injective functions.

要判断一个函数是否有反函数，可以使用水平线检验：当且仅当任意一条水平线与函数图像最多相交于一个点时，该函数才有反函数。这样的函数称为一对一函数，或单射函数。

For example, y = 2x + 3 passes the horizontal line test because every horizontal line meets it exactly once. However, y = x² without a domain restriction fails the test, because a horizontal line such as y = 4 meets the graph at x = -2 and x = 2.

例如，y = 2x + 3 能通过水平线检验，因为每条水平线都与它相交于一点。然而，没有定义域限制的 y = x² 无法通过检验，因为水平线 y = 4 会与图像相交于 x = -2 和 x = 2 两点。

If a function fails the horizontal line test, we can often restrict its domain to make it one-to-one. That is exactly what we did with f(x) = x², x ≥ 0.

如果一个函数无法通过水平线检验，我们通常可以通过限制定义域使它成为一对一函数。这正是我们对 f(x) = x², x ≥ 0 所做的处理。

9. Composite Functions and Inverses | 反函数与复合函数

Composing a function with its inverse in either order gives the identity function. This is one of the most useful checking methods in the exam.

将函数与其反函数按任意顺序复合，都会得到恒等函数。这是考试中最实用的验算方法之一。

f(f⁻¹(x)) = x and f⁻¹(f(x)) = x

Using f(x) = 2x + 3 and f⁻¹(x) = (x – 3)/2, let us verify.

以 f(x) = 2x + 3 和 f⁻¹(x) = (x – 3)/2 为例，我们来验证。

f⁻¹(f(x)) = f⁻¹(2x + 3) = ((2x + 3) – 3)/2 = x

f(f⁻¹(x)) = f((x – 3)/2) = 2 × ((x – 3)/2) + 3 = x

Both compositions simplify to x, confirming that the two functions are true inverses of each other.

两个复合结果都化简为 x，这就确认了这两个函数互为反函数。

10. Common Mistakes and Exam Tips | 常见错误与考试提示

Many students lose marks on inverse function questions because of small but avoidable errors. Keep the following points in mind during your revision and in the exam.

许多学生在反函数题目上因为一些小而可以避免的错误而失分。在复习和考试时，请务必牢记以下要点。

Do not confuse f⁻¹(x) with 1/f(x). The symbol f⁻¹ means the inverse function, not the reciprocal.

不要把 f⁻¹(x) 与 1/f(x) 混淆。符号 f⁻¹ 表示反函数，而不是倒数。
Always start by writing y = f(x) before swapping variables. Skipping this step often leads to sign errors.

在交换变量之前，一定要先写出 y = f(x)。跳过这一步常常会导致符号错误。
Check the domain and range after finding the inverse. The range of f becomes the domain of f⁻¹.

求出反函数后要检查定义域和值域。f 的值域会变成 f⁻¹ 的定义域。
Remember that a function must be one-to-one to have an inverse. If it is not, restrict the domain.

记住，函数必须是一对一才能有反函数。如果不是，就要限制定义域。
When drawing the graph of an inverse, reflect the original graph in the line y = x and label key coordinates.

画反函数图像时，要将原图像关于直线 y = x 作对称，并标出关键坐标。

If you use the composition check before writing your final answer, you can eliminate most careless errors and gain confidence in your result.

如果在写出最终答案之前使用复合运算进行检验，你就可以消除大部分粗心错误，并对结果更有把握。

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Inverse Functions | 反函数