Exercise F: Composite and Inverse Functions | 练习F：复合函数与反函数

📚 Exercise F: Composite and Inverse Functions | 练习F：复合函数与反函数

In IB Mathematics, one of the most fundamental and frequently examined sets of skills is the manipulation of functions, specifically creating composite functions and finding inverses. This exercise set, commonly labelled as 'Exercise F' in many standard textbooks, consolidates your understanding of how functions can be combined, reversed, and interpreted both algebraically and graphically. Mastering these concepts is not just about passing a test; it underpins topics in calculus, transformation geometry, and modelling real-world relationships.

在IB数学中，最基本且常考的技能之一就是对函数进行运算，尤其是构造复合函数和求解反函数。这套练习在许多标准教材中通常被标为“练习F”，旨在帮助你巩固对函数如何组合、逆转以及从代数和图形两方面加以理解的掌握。掌握这些概念不仅是为了通过考试，更是微积分、变换几何和现实关系建模等主题的基础。

1. What Is a Composite Function? | 什么是复合函数？

A composite function is created when one function is applied to the result of another function. Given two functions f(x) and g(x), the composite function f(g(x)) means you first put x into g, get an output, and then put that output into f. This notation is read as 'f of g of x' or 'f composed with g'. The order is crucial: f(g(x)) is generally different from g(f(x)), unless the functions are special inverses of one another.

复合函数是指将一个函数作用于另一个函数的结果所形成的新函数。给定两个函数 f(x) 和 g(x)，复合函数 f(g(x)) 表示先将 x 代入 g，得到输出，再将该输出代入 f。这个记法读作“f of g of x”或“f 与 g 的复合”。顺序至关重要：除非两函数互为特殊的反函数，否则 f(g(x)) 通常与 g(f(x)) 不同。

2. Notation and Order of Composition | 复合的记号与顺序

In IB exam papers, you will see both f(g(x)) and (f ∘ g)(x) to denote the same composite function. The symbol ∘ is the composition operator, and (f ∘ g)(x) is evaluated exactly as f(g(x)). Notice that the function on the right, g, is applied first. A common mistake is to read f ∘ g and apply f first. Remember: the function closest to the x is the inner function and acts first. Always rewrite (f ∘ g)(x) as f(g(x)) to avoid confusion.

在IB试卷中，你会看到 f(g(x)) 和 (f ∘ g)(x) 两种记号表示同样的复合函数。符号 ∘ 是复合运算符，(f ∘ g)(x) 的计算完全等同于 f(g(x))。注意，右边的函数 g 首先被应用。常见错误是看到 f ∘ g 却先应用 f。记住：最靠近 x 的函数是内层函数，最先执行。为避免混淆，始终将 (f ∘ g)(x) 改写为 f(g(x))。

3. Domain and Range of Composite Functions | 复合函数的定义域与值域

Determining the domain of a composite function f(g(x)) requires a two-step check. First, x must be in the domain of the inner function g. Second, the output g(x) must be in the domain of the outer function f. The composite domain is therefore the set of all x in the domain of g such that g(x) is in the domain of f. In contrast, finding the range is often done by first finding the range of the inner function over the restricted domain and then applying the outer function to that set.

确定复合函数 f(g(x)) 的定义域需要两步检查。首先，x 必须在内部函数 g 的定义域内。其次，输出 g(x) 必须在外部函数 f 的定义域内。因此，复合定义域是 g 的定义域中使得 g(x) 属于 f 的定义域的所有 x 的集合。相反，求值域通常先求出内部函数在限定定义域下的值域，然后再将外部函数应用于该集合。

4. Worked Example on Composite Functions | 复合函数例题解析

Let f(x) = √x for x ≥ 0 and g(x) = x − 3 for all real x. Find (f ∘ g)(x) and its domain. First, (f ∘ g)(x) = f(g(x)) = √x − 3 this to be defined, the expression inside the square root must be non-negative: x − 3 ≥ 0 ⇒ x − 3 is defined for all real x, so the only restriction comes from f. Hence, domain is {x | x ≥ 3}. Note that the range is y ≥ 0 because square root outputs are non-negative.

设 f(x) = √x，x ≥ 0，且 g(x) = x − 3，x 为所有实数。求 (f ∘ g)(x) 及其定义域。首先，(f ∘ g)(x) = f(g(x)) = √x − 3。要使此式有意义，平方根内的表达式必须非负：x − 3 ≥ 0 ⇒ x ≥ 3。同时，g(x) = x − 3 对所有实数有定义，因此限制仅来自 f。所以，定义域为 {x | x ≥ 3}。注意值域为 y ≥ 0，因为平方根输出非负。

5. Introduction to Inverse Functions | 反函数简介

An inverse function, denoted f⁻¹(x), essentially 'undoes' range of f. It is critical to understand that the superscript −1 is not an exponent; f⁻¹(x) does not mean 1/f(x).

反函数，记作 f⁻¹(x)，本质上是“撤销”原函数 f(x) 所做操作的一种函数。如果 f 将 a 映射到 b，那么 f⁻¹ 将 b 映射回 a。这意味着先应用 f 再应用 f⁻¹ 会返回原始输入：对于 f 定义域内的所有 x，有 f⁻¹(f(x)) = x。类似地，对于 f 值域内的所有 x，有 f(f⁻¹(x)) = x。至关重要的是要理解上标 −1 不是指数；f⁻¹(x) 不表示 1/f(x)。

6. When Does an Inverse Function Exist? | 反函数存在的条件

on is not one-to-one over its entire domain, and therefore does not have an inverse unless we restrict the domain appropriately. For instance, f(x) = x² fails the horizontal line test on all real numbers, but if we restrict the domain to x ≥ 0, it becomes invertible with inverse f⁻¹(x) = √x.

并非每个函数都有反函数。一个函数要有反函数，它必须是一一对应的（单射），这意味着不存在两个不同的输入产生相同的输出。从图形上看，可通过水平线检验：如果任意水平线与图像相交多于一次，则该函数在其整个定义域上不是一一对应，因而没有反函数，除非我们适当限制定义域。例如，f(x) = x² 在所有实数上不能通过水平线检验，但如果将定义域限制为 x ≥ 0，它就变得可逆，其反函数为 f⁻¹(x) = √x。

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7. Finding Inverse Functions Algebraically | 代数求解反函数

r example, if f(x) = (2x + 1)/(x − 3), setting y = (2x + 1)/(x − 3), swapping gives x = (2y + 1)/(y − 3). Solving yields y = (3x + 1)/(x − 2), so f⁻¹(x) = (3x + 1)/(x − 2), with domain x ≠ 2 (since x=2 makes denominator zero, and original range excludes 2).

要用代数方法求函数 f(x) 的反函数，请遵循以下标准步骤：1) 写出 y = f(x)。2) 交换 x 和 y 得到 x = f(y)。3) 解出这个新方程中关于 x 的 y。4) 所得表达式即为 y = f⁻¹(x)。始终要标明 f⁻¹ 的定义域，即原函数 f 的值域。例如，若 f(x) = (2x + 1)/(x − 3)，设 y = (2x + 1)/(x − 3)，交换后得 x = (2y + 1)/(y − 3)。解出 y = (3x + 1)/(x − 2)，故 f⁻¹(x) = (3x + 1)/(x − 2)，定义域为 x ≠ 2（因为 x=2 使分母为零，且原函数值域排除 2）。

8. Graph of a Function and Its Inverse | 函数及其反函数的图像

s means that if a point (a, b) lies on the graph of f, then the point (b, a) lies on the graph of f⁻¹. This reflection property is extremely useful for sketching inverses quickly and for understanding the symmetry between domain and range. It also visually confirms the horizontal line test: if the original function is one-to-one, its reflection will pass the vertical line test, confirming it is a function.

有一个强大的性质将 f 与 f⁻¹ 的图像联系起来：它们关于直线 y = x 互为镜像。这意味着，若点 (a, b) 在 f 的图像上，则点 (b, a) 就在 f⁻¹ 的图像上。这一反射性质对于快速绘制反函数草图以及理解定义域与值域之间的对称性极为有用。它还能在视觉上验证水平线检验：如果原函数是一一对应的，其反射图像就能通过垂直线检验，从而确认它是一个函数。

9. The Relationship Between Composite and Inverse Functions | 复合函数与反函数的关系

inition, f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. In composition notation, this is written as (f⁻¹ ∘ f)(x) = x and (f ∘ en f and g are inverse functions. This relationship is a favourite in IB exam questions, where you might be asked to show that two given functions are inverses using composition.

复合函数与反函数的定义密切相关。根据定义，f⁻¹(f(x)) = x 且 f(f⁻¹(x)) = x。用复合记号可写作 (f⁻¹ ∘ f)(x) = x 和 (f ∘ f⁻¹)(x) = x。这个性质常被用来验证两个函数是否互为反函数：如果在各自定义域内对所有 x 都有 f(g(x)) = x 和 g(f(x)) = x，那么 f 和 g 就是反函数。这种关系是IB考试中的热门考点，题目可能会要求你利用复合运算证明两个给定函数互为反函数。

10. Common Pitfalls and Misconceptions | 常见误区

Students often mistakenly treat f⁻¹(x) as 1/f(x). This is disastrous in exams. Another typical error is forgetting to check or state the domain of the inverse. Also, when composing, students may apply the functions in the wrong order, especially when using the ∘ notation. Moreover, assuming a function has an inverse without checking one-to-one property can lead to incorrect results, particularly with quadratics or trigonometric functions unless a domain restriction is specified. Always check the horizontal line test.

学生经常错误地将 f⁻¹(x) 视为 1/f(x)，这在考试中是致命的。另一个典型错误是忘记检查或说明反函数的定义域。此外，在进行复合运算时，学生可能搞错函数的应用顺序，尤其是在使用 ∘ 记号时。再者，不经检查一一对应性质就假设函数有反函数会导致错误结果，尤其对于二次函数或三角函数，除非明确指定了定义域限制，否则不能求反函数。务必进行水平线检验。

11. Typical IB Question Styles for This Exercise | 练习中常见的IB题型

contexts. You might be asked: given f(x) = 2x + 1 and g(x) = x² − 3, find f(g(2)), then (g ∘ f)(x). Or, find the inverse of h(x) = e^2x+1 on may involve a rational function where you must find its inverse and then verify that (h ∘ y = f(x) and y = f⁻¹(x) on the same axes, identifying their intersection points on y = x.

练习F的题目通常从基本的复合运算逐步过渡到更具挑战性的情境。你可能会被问到：已知 f(x) = 2x + 1 和 g(x) = x² − 3，求 f(g(2))，然后求 (g ∘ f)(x)。或者求 h(x) = e^2x+1 的反函数并指明其定义域。较难的题目可能涉及一个有理函数，要求求出其反函数然后验证 (h ∘ h⁻¹)(5) = 5。图形解释题会要求你在同一坐标系中画出 y = f(x) 和 y = f⁻¹(x) 的草图，并找出它们在 y = x 上的交点。

12. Summary and Final Tips | 总结与终极建议

heck your graphs mentally. When practising Exercise F, work systematically through each step and verify your answers by composition. These foundations will reappear throughout your IB course, especially in calculus when dealing with the chain rule and integration by substitution.

复合函数与反函数构成了一套要求既能熟练代数运算又需概念清晰的综合技能。始终遵循顺序：先内层函数。对于反函数，交换 x 和 y 后求解，且永远不要忘记定义域限制。利用关于 y = x 的反射性质在头脑中审视图像。在练习F时，按步骤有条不紊地操作，并通过复合运算验证你的答案。这些基础将贯穿你的整个IB课程，尤其是在微积分中处理链式法则和换元积分时会反复出现。

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