📚 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？
s 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)
= 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。
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Write y = f(x).
写出 y = f(x)。
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Swap x and y in the equation.
将方程中的 x 与 y 互换。
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Rearrange the equation to make y the subject.
重新整理方程，使 y 成为主项。
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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。
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Step 1: Write y = 2x + 3.
第一步：写出 y = 2x + 3。
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Step 2: Swap x and y: x = 2y + 3.
第二步：交换 x 和 y：x = 2y + 3。
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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
is 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
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。
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Step 1: Write y = x².
第一步：写出 y = x²。
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Step 2: Swap x and y: x = y².
第二步：交换 x 和 y：x = y²。
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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
≥ 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
f 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
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。
ils 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。
trict 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。
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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。
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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。
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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。
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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。
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aph 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。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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