📚 G - Inverse functions | 逆函数
An inverse function reverses the operation of a given function. If a function f maps x to y, then its inverse f⁻¹ maps y back to x, provided the original function is one-to-one. This concept is fundamental in IB Mathematics, as it connects algebraic manipulation, graphical interpretation, and calculus.
逆函数会反转原函数的运算。如果一个函数 f 将 x 映射到 y，那么它的逆 f⁻¹ 会将 y 映回 x，前提是原函数是一一对应的。这一概念在 IB 数学中十分基本，因为它将代数操作、图象解释与微积分联系起来。
1. What is an Inverse Function? | 什么是逆函数？
ion undoes the original: applying f then f⁻¹ brings you back to the starting value, and vice versa.
若函数 f 的定义域为 X、值域为 Y，且对 Y 中每个 y，在 X 中恰有一个 x 满足 f(x) = y，则存在逆函数 f⁻¹，并定义 f⁻¹(y) = x。逆运算会抵消原运算：先施加 f 再施加 f⁻¹ 将回到初始值，反之亦然。
2. One-to-One Functions and the Horizontal Line Test | 一一函数与水平线检验
For a function to have an inverse that is itself a function, it must be one-to-one (injective). This means that no two different inputs produce the same output. Graphically, if any horizontal line cuts the graph more than once, the function is not one-to-one and does not have an inverse function over its entire domain.
函数要有也是函数的逆，它必须是一一函数（单射）。这意味着没有两个不同的输入会产生相同的输出。从图象上看，若任意一条水平线与图象相交多于一次，则该函数不是一一的，在其整个定义域上不存在逆函数。
3. Domain and Range of Inverse Functions | 逆函数的定义域与值域
The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f. This swapping of domain and range is a key property. Whenever you find an inverse, you must state its domain explicitly, often restricted from the original function's range.
逆函数 f⁻¹ 的定义域是原函数 f 的值域，f⁻¹ 的值域则是 f 的定义域。定义域与值域的互换是一个关键性质。每次求逆函数时，都必须明确写出其定义域，它通常受到原函数值域的约束。
4. Finding the Inverse Function Algebraically | 用代数法求逆函数
y = f⁻¹(x). Finally, state the domain of f⁻¹ based on the range of the original function.
用代数方法求逆函数的步骤如下：(1) 将函数写成 y = f(x)。(2) 交换 x 和 y，得到 x = f(y)。(3) 解出 y 关于 x 的表达式。(4) 所得结果就是 y = f⁻¹(x)。最后，根据原函数的值域确定 f⁻¹ 的定义域。
5. Graphical Relationship: Symmetry about y = x | 图象关系：关于 y = x 对称
is is because swapping x and y geometrically mirrors every point (a, b) to (b, a). You can use this symmetry to sketch the inverse once the original graph is known.
f 与 f⁻¹ 的图象关于直线 y = x 对称。这是因为交换 x 与 y 在几何上会将每一点 (a, b) 映射为 (b, a)。知道原函数的图象后，即可利用这一对称性画出逆函数的图象。
6. Notation: f⁻¹(x) vs. [f(x)]⁻¹ | 符号：f⁻¹(x) 与 [f(x)]⁻¹ 的区别
Be careful: f⁻¹(x) denotes the inverse function, not the reciprocal. The reciprocal is written as [f(x)]⁻¹ or 1/f(x). For trigonometric functions, sin⁻¹ x means the inverse sine, while (sin x)⁻¹ is the reciprocal cosecant. Always interpret the superscript -1 in context.
请注意：f⁻¹(x) 表示逆函数，而不是倒数。倒数写作 [f(x)]⁻¹ 或 1/f(x)。对于三角函数，sin⁻¹ x 表示反正弦，而 (sin x)⁻¹ 则是余割。始终要根据上下文理解上标 -1 的含义。
7. Restricting the Domain to Obtain an Inverse | 限制定义域以获得逆函数
that does have an inverse. For example, f(x) = x² restricted to x ≥ 0 has inverse f⁻¹(x) = √x. The choice of domain branch must be clearly stated.
许多函数（如二次函数）在其自然定义域上不是一一的。通过限制定义域，我们可以构造一个一一函数，使之存在逆函数。例如，将 f(x) = x² 限制于 x ≥ 0，其逆为 f⁻¹(x) = √x。必须清楚地说明所选的定义域分支。
8. Inverse of Linear, Rational and Polynomial Functions | 线性、有理与多项式函数的逆
)/(cx + d) also have inverses provided ad - bc ≠ 0; the inverse can be found by cross-multiplying and rearranging. Higher-degree polynomials require domain restriction and often square or cube roots.
对于线性函数 f(x) = ax + b，通过解 x = ay + b 得到 y，即 f⁻¹(x) = (x - b)/a (a ≠ 0)。有理函数如 f(x) = (ax + b)/(cx + d) 在 ad - bc ≠ 0 时也有逆函数，可通过交叉相乘和移项求出。高次多项式则需要限制定义域，并常涉及平方根或立方根。
9. Exponential and Logarithmic Functions as Inverses | 指数函数与对数函数互为逆
The exponential function f(x) = aˣ > 0, a ≠ 1) and the logarithmic function g(x) = logₐ x are inverses of each other. Thus, alogₐ x = x and logₐ (aˣ) = x. The natural exponential eˣ x follow the same relationship, which is vital for solving equations and modelling growth.
指数函数 f(x) = aˣ (a > 0, a ≠ 1) 与对数函数 g(x) = logₐ x 互为逆函数。因此有 alogₐ x = x 和 logₐ (aˣ) = x。自然指数 eˣ 与自然对数 ln x 也满足同一关系，这在解方程和增长建模中至关重要。
10. Inverse Trigonometric Functions | 反三角函数
x has range (-π/2, π/2). These inverses allow us to solve equations like sin θ = k for θ.
三角函数是周期函数，不是一一的，因此必须限制定义域以定义反三角函数。对于正弦，限制在 [-π/2, π/2] 上得到 arcsin x 或 sin⁻¹ x。类似地，arccos x 的值域为 [0, π]，arctan x 的值域为 (-π/2, π/2)。这些反函数让我们能够求解诸如 sin θ = k 的方程。
11. Composition of a Function and Its Inverse | 函数与其逆的复合
his property can be used to verify that two functions are indeed inverses. Simply compute both compositions and check that they simplify to the identity function.
对于函数 f 及其逆 f⁻¹，复合满足 f(f⁻¹(x)) = x（对所有 f⁻¹ 定义域中的 x）和 f⁻¹(f(x)) = x（对所有 f 定义域中的 x）。这一性质可用来验证两个函数是否互为逆函数。只需计算两个复合式，检查它们是否化简为恒等函数即可。
12. Derivative of an Inverse Function (HL) | 逆函数的导数（高水平）
function can be found without explicitly knowing the inverse. If f is differentiable and f'(f⁻¹(x)) ≠ 0, then (f⁻¹)'(x) = 1 / f' ⁻¹(x)) = x. It is particularly useful when the inverse is difficult to express explicitly.
对于 HL 学生，逆函数的导数可以在不显式求出逆函数的情况下求得。若 f 可微且 f'(f⁻¹(x)) ≠ 0，则 (f⁻¹)'(x) = 1 / f'(f⁻¹(x))。该公式由对 f(f⁻¹(x)) = x 应用链式法则推导而来。在逆函数难以显式表达时，这一公式尤为有用。
(f⁻¹)'(x) = 1 / f'( f⁻¹(x) )
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