Inverse Functions | 反函数

📚 Inverse Functions | 反函数

An inverse function reverses the effect of a function. If a function f sends an input x to an output f(x), then the inverse function f⁻¹ sends f(x) back to x. Understanding inverses is essential for solving equations, analysing graphs and working with exponential, logarithmic and trigonometric functions.

反函数的作用是逆转原函数的效果。如果函数 f 将输入 x 映射到输出 f(x)，那么反函数 f⁻¹ 就会将 f(x) 还原为 x。理解反函数对于解方程、分析图像以及处理指数、对数和三角函数都至关重要。

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

For a function f, its inverse f⁻¹ is defined by the property that for every x in the domain of f, f⁻¹(f(x)) = x, and for every y in the domain of f⁻¹, f(f⁻¹(y)) = y. In other words, applying f and then f⁻¹ brings you back to your starting value.

对于函数 f，其反函数 f⁻¹ 满足：对 f 定义域中的每一个 x，都有 f⁻¹(f(x)) = x；对 f⁻¹ 定义域中的每一个 y，都有 f(f⁻¹(y)) = y。也就是说，先应用 f 再应用 f⁻¹，就会回到最初的取值。

Not every function has an inverse. The function must be one-to-one (injective), meaning that each output value comes from exactly one input value. If two different inputs produce the same output, we cannot reverse the process uniquely.

并非每个函数都有反函数。函数必须是一一对应的（单射），这意味着每个输出值只能来自唯一的输入值。如果两个不同的输入产生了相同的输出，就无法唯一地逆转这个过程。

f⁻¹(f(x)) = x and f(f⁻¹(y)) = y

2. The Condition for an Inverse: One-to-One Functions | 反函数存在的条件：一一对应函数

A function f is one-to-one if, whenever f(a) = f(b), we must have a = b. Graphically, a one-to-one function passes the horizontal line test: any horizontal line intersects the graph at most once.

函数 f 是一一对应的，如果当 f(a) = f(b) 时，必然有 a = b。从图像上看，一一对应函数满足水平线检验：任何水平直线与图像的交点最多只有一个。

If a horizontal line crosses the graph more than once, the function is not one-to-one and has no inverse over its whole domain.

如果一条水平直线与图像相交超过一次，该函数就不是一一对应的，因而在其整个定义域上没有反函数。
We can sometimes restrict the domain to make a function one-to-one. For example, f(x) = x² is not one-to-one on ℝ, but on x ≥ 0 it becomes one-to-one.

有时我们可以通过限制定义域使函数成为一一对应。例如，f(x) = x² 在 ℝ 上不是一一对应的，但在 x ≥ 0 上它是一一对应的。

3. Finding the Inverse of a Function | 求函数的反函数

To find the inverse of a function f(x):

求函数 f(x) 的反函数的步骤：

Write the function as y = f(x).

将函数写成 y = f(x)。
Swap x and y.

交换 x 和 y。
Rearrange to make y the subject.

重新整理，使 y 成为被表示的对象。
Replace y with f⁻¹(x).

将 y 替换为 f⁻¹(x)。

For example, let f(x) = 3x + 2. Write y = 3x + 2, swap to get x = 3y + 2, then solve: y = (x − 2) / 3. Thus f⁻¹(x) = (x − 2) / 3.

例如，设 f(x) = 3x + 2。先写 y = 3x + 2，交换后得到 x = 3y + 2，然后解出 y = (x − 2) / 3。因此 f⁻¹(x) = (x − 2) / 3。

4. Domain and Range of the Inverse | 反函数的定义域和值域

An important relationship: the domain of f⁻¹ is exactly the range of f, and the range of f⁻¹ is exactly the domain of f.

一个重要关系：f⁻¹ 的定义域恰好是 f 的值域，而 f⁻¹ 的值域恰好是 f 的定义域。

Domain(f⁻¹) = Range(f), Range(f⁻¹) = Domain(f)

When you find an inverse algebraically, always check the domain and range. For instance, if f(x) = x² for x ≥ 0, then f⁻¹(x) = √x, with domain x ≥ 0 and range y ≥ 0.

当你通过代数方法求出反函数时，一定要检查定义域和值域。例如，若 f(x) = x²（x ≥ 0），则 f⁻¹(x) = √x，其定义域为 x ≥ 0，值域为 y ≥ 0。

In AQA exams, marks are often awarded for stating the domain or range of an inverse function. Do not forget to include them.

在 AQA 考试中，写清反函数的定义域或值域常常是给分点。千万不要遗漏。

5. Graph of an Inverse Function | 反函数的图像

The graph of f⁻¹ is the reflection of the graph of f in the line y = x. This is because swapping x and y is equivalent to reflecting across that diagonal line.

函数 f⁻¹ 的图像是 f 的图像关于直线 y = x 的对称图像。因为交换 x 和 y 就相当于沿这条对角线反射。

To sketch the inverse, you can reflect a few key points. If (a, b) lies on y = f(x), then (b, a) lies on y = f⁻¹(x).

要画出反函数的图像，你可以反射几个关键点。如果 (a, b) 在 y = f(x) 上，那么 (b, a) 就在 y = f⁻¹(x) 上。

(a, b) on f ⟷ (b, a) on f⁻¹

Note that a function and its inverse intersect wherever they meet the line y = x, and if they intersect elsewhere, those points come in symmetric pairs.

注意，函数与其反函数在它们与直线 y = x 相交的地方相交；如果它们在其他地方相交，这些交点会成对称对出现。

6. Composite Functions and Inverses | 复合函数与反函数

The composition of a function and its inverse gives the identity function. For a function f with inverse f⁻¹:

函数与其反函数的复合得到恒等函数。对于函数 f 及其反函数 f⁻¹：

f⁻¹ ∘ f (x) = x and f ∘ f⁻¹ (x) = x

This property is useful for solving equations. For example, to solve √x = 5, apply the inverse of the square root function (squaring) to both sides: x = 25.

这一性质在解方程时非常有用。例如，要解 √x = 5，可以对两边应用平方根函数的反函数（平方）：x = 25。

When dealing with composite functions like f(g(x)), the inverse satisfies (f ∘ g)⁻¹ = g⁻¹ ∘ f⁻¹. Note the order reverses, just as when undressing you take off the coat before the shirt.

处理像 f(g(x)) 这样的复合函数时，其反函数满足 (f ∘ g)⁻¹ = g⁻¹ ∘ f⁻¹。注意顺序会颠倒，就像脱衣服时先脱外套再脱衬衫一样。

7. Self-Inverse Functions | 自反函数

A function is self-inverse if f⁻¹(x) = f(x) for all x in its domain. This means that applying the function twice returns the original input: f(f(x)) = x.

如果对于定义域中的所有 x，都有 f⁻¹(x) = f(x)，则该函数是自反函数。这意味着对输入应用两次该函数后会回到原始输入：f(f(x)) = x。

Classic examples include f(x) = 1/x (for x ≠ 0) and f(x) = −x. Graphically, a self-inverse function is symmetric about the line y = x.

典型的例子包括 f(x) = 1/x（x ≠ 0）和 f(x) = −x。从图像上看，自反函数关于直线 y = x 对称。

Recognising self-inverse functions can save time in exams, as you immediately know that the inverse has the same expression as the original function.

识别自反函数可以在考试中节省时间，因为你立刻知道反函数与原函数具有相同的表达式。

8. Inverse Trigonometric Functions | 反三角函数

Trigonometric functions like sin x, cos x and tan x are periodic, so they are not one-to-one on their natural domains. To define inverses, we restrict their domains:

三角函数的 sin x、cos x 和 tan x 是周期性的，因此它们在其自然定义域上不是一一对应的。为了定义反函数，我们需要限制它们的定义域：

sin⁻¹ x (or arcsin x) has domain −1 ≤ x ≤ 1 and range −π/2 ≤ y ≤ π/2.

sin⁻¹ x（或 arcsin x）的定义域为 −1 ≤ x ≤ 1，值域为 −π/2 ≤ y ≤ π/2。
cos⁻¹ x (or arccos x) has domain −1 ≤ x ≤ 1 and range 0 ≤ y ≤ π.

cos⁻¹ x（或 arccos x）的定义域为 −1 ≤ x ≤ 1，值域为 0 ≤ y ≤ π。
tan⁻¹ x (or arctan x) has domain all real numbers and range −π/2 < y < π/2.

tan⁻¹ x（或 arctan x）的定义域为全体实数，值域为 −π/2 < y < π/2。

Remember that sin⁻¹ x is not the same as (sin x)⁻¹ = 1/sin x. The notation sin⁻¹ x means the inverse function, not the reciprocal.

请记住，sin⁻¹ x 与 (sin x)⁻¹ = 1/sin x 不同。记号 sin⁻¹ x 表示反函数，而不是倒数。

9. Exponential and Logarithmic Inverses | 指数函数与对数函数的互逆

The exponential function aˣ and the logarithmic function logₐ x are inverse functions of each other, provided a > 0 and a ≠ 1.

指数函数 aˣ 和对数函数 logₐ x 互为反函数，前提是 a > 0 且 a ≠ 1。

logₐ(aˣ) = x and a^(logₐ x) = x

In particular, eˣ and ln x (natural logarithm) are inverses. This relationship is used constantly in solving exponential and logarithmic equations.

特别地，eˣ 与 ln x（自然对数）互为反函数。这一关系在解指数和对数方程时经常被使用。

For example, to solve eˣ = 7, take the natural logarithm of both sides: x = ln 7. Conversely, to solve ln x = 3, exponentiate both sides: x = e³.

例如，要解 eˣ = 7，对两边取自然对数：x = ln 7。反过来，要解 ln x = 3，对两边取指数：x = e³。

When differentiating or integrating, remembering that eˣ and ln x are inverses helps simplify expressions like e^(ln x) = x.

在求导或积分时，记住 eˣ 与 ln x 互为反函数有助于简化像 e^(ln x) = x 这样的表达式。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Students often make the same mistakes when dealing with inverse functions. Being aware of these can help you secure full marks.

学生在处理反函数时常犯一些相同的错误。了解这些错误可以帮助你拿到满分。

Do not confuse f⁻¹(x) with (f(x))⁻¹. The former is the inverse function; the latter is the reciprocal 1/f(x).

不要把 f⁻¹(x) 与 (f(x))⁻¹ 混淆。前者是反函数，后者是倒数 1/f(x)。
Always state the domain of the inverse. Since the range of f must be found first, this often requires considering the turning points of f.

始终标明反函数的定义域。因为需要先求出 f 的值域，这通常要考虑 f 的极值点。
Check that your inverse is correct by verifying f⁻¹(f(x)) = x for a few values of x.

通过代入一些 x 值验证 f⁻¹(f(x)) = x 是否成立，来检查反函数是否正确。
When a function is not one-to-one, specify a restricted domain before attempting to find the inverse.

当函数不是一一对应时，在尝试求反函数之前，需要先指定一个受限的定义域。
For inverse trigonometric functions, use the restricted principal value ranges shown in section 8.

对于反三角函数，使用第 8 节中给出的受限主值范围。

In AQA A-Level Mathematics, inverse function questions often appear as part of larger problems involving transformations, graphs or solving equations. Practice sketching reflections in the line y = x and always pay close attention to domains.

在 AQA A-Level 数学中，反函数问题常常作为变换、图像或解方程等更大问题的一部分出现。练习画关于直线 y = x 的反射图像，并且始终密切关注定义域。

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Inverse Functions | 反函数