📚 Exercise 4D.2: Composite Functions | 练习4D.2：复合函数
Exercise 4D.2 in the IB Mathematics syllabus is entirely dedicated to building confidence with composite functions. These appear when one function is applied to the output of another, and they form the backbone of later topics such as inverse functions, transformations, and the chain rule in calculus. This article walks you through the key ideas, worked examples, common pitfalls, and exam strategies so you can handle every variation of the exercise with ease.
IB数学大纲中的练习4D.2完全旨在建立对复合函数的信心。当一个函数作用于另一个函数的输出时，就产生了复合函数，它们是反函数、变换以及微积分中链式法则等后续主题的基础。本文将带你梳理核心概念、详细例题、常见错误和考试策略，使你能够轻松应对该练习中的所有变化。
1. What is a Composite Function? | 什么是复合函数？
A composite function is formed by linking two functions so that the output of the first becomes the input of the second. For two functions f and g, the composite f(g(x)) – read as 'f of g of x' – means you apply g to x first, then apply f to the result. The process is like a two‑step machine: x enters g, an intermediate value comes out, and that value goes straight into f.
复合函数是将两个函数连接起来，使第一个函数的输出成为第二个函数的输入。对于两个函数f和g，复合函数f(g(x))——读作“f of g of x”——意味着先将g应用于x，然后将f应用于得到的结果。这个过程就像一台两步机器：x进入g，输出一个中间值，该值直接进入f。
Understanding this order is the single most important thing. The notation (f∘g)(x) is simply another way of writing f(g(x)). Always remember that the function closest to the variable is performed first, just as when you dress yourself you put on socks before shoes – reversing the order gives a completely different result.
理解这个顺序是最重要的事。符号(f∘g)(x)只是f(g(x))的另一种写法。永远记住：最靠近变量的函数最先被执行，就像你穿衣服时先穿袜子再穿鞋子——颠倒顺序会得到完全不同的结果。
2. Notation: (f∘g)(x) and f(g(x)) | 符号：(f∘g)(x) 与 f(g(x))
IB exam papers use both f(g(x)) and (f∘g)(x) interchangeably. The small circle '∘' stands for composition, and you will often see expressions like f∘g or h = f∘ that fg(x) without the circle is ambiguous and should be avoided in your written solutions – always include the brackets or the composition symbol to make your intention clear.
IB试卷中交替使用f(g(x))和(f∘g)(x)。小圆圈“∘”代表复合运算，你经常会看到诸如f∘g或h = f∘g的表达式。请注意，没有圆圈的fg(x)含义模糊，书写答案时应避免——务必加上括号或复合符号以使意图清晰。
When the question asks for (f∘g)(x), write f(g(x)) and proceed with the substitution. The two notations are treated as identical, and using either will earn full marks as long as the algebraic work is correct.
当题目要求(f∘g)(x)时，写成f(g(x))并代入运算即可。这两种记法被视为完全等价，只要代数运算正确，使用任何一种都能得满分。
3. Step-by-Step Evaluation | 逐步求值方法
To evaluate a composite function such as f(g(x)), follow a simple recipe: write down the expression for g(x), then substitute that entire expression wherever the variable appears in f. Finally, simplify by expanding brackets and collecting like terms. If you are evaluating at a number, you can either simplify the composite first or work from the inside out.
要计算诸如f(g(x))的复合函数，只需遵循简单的步骤：先写下g(x)的表达式，然后将整个表达式替换f中所有出现变量的地方。最后，通过展开括号和合并同类项来化简。如果是求数值，你可以先化简复合函数，也可以从内层向外层逐步计算。
For f(x)=2x+3 and g(x)=x²−1, f(g(x)) = 2(x²−1)+3 = 2x²+1
对于 f(x)=2x+3 和 g(x)=x²−1，f(g(x)) = 2(x²−1)+3 = 2x²+1
When working with fractions or radicals, extra care is needed. Write every substitution inside parentheses – even if it seems obvious – so that negative signs and exponents are handled correctly. This habit will save you from sign errors and lost marks.
处理分式或根式时需格外小心。把每一步的代入都放在括号内——即使看起来很明显——这样能正确处理负号和指数。这个习惯将避免符号错误和失分。
4. Domain and Range Considerations | 定义域与值域的注意事项
The domain of f(g(x)) consists of those x‑values that belong to the domain of g and for which g(x) lies in the domain of f. This means you cannot just look at the final simplified formula – you must trace the restrictions backward through both functions.
f(g(x))的定义域由那些属于g的定义域且使得g(x)落在f的定义域内的x值组成。这意味着你不能只看最终化简后的表达式——必须逆向追踪两个函数的限制条件。
For instance, if f(x)=√x and g(x)=x−4, then f(g(x))=√x−4 issue, but the outer square root imposes a restriction. If instead g(x)=1/(x−2) and f(x)=√x xclude x=2 from the domain because g is undefined there, on top of the radicand condition.
例如，若f(x
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