📚 Composite Functions | 复合函数考点精讲
In GCSE WJEC Mathematics, understanding composite functions is essential for mastering algebra and function notation. This topic builds on basic function skills, requiring you to combine two or more functions into a single expression and evaluate them correctly. Whether you are substituting numbers or manipulating algebraic formulas, a clear grasp of how to form and interpret f(g(x)) directly impacts your exam performance. Below, we unpack every key point, from first principles to exam-style challenges, so you can approach composite functions with confidence.
在 GCSE WJEC 数学中，理解复合函数是掌握代数与函数记法的关键。这部分内容建立在基本函数技能之上，要求你将两个或多个函数组合成一个表达式并正确求值。无论是代入数值还是处理代数式，能否清晰构建并解读 f(g(x)) 直接决定了你的考试表现。下面我们将从基本原理到考试型难题逐一剖析，帮助你自信应对复合函数。
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 output of another function. It can be thought of as a two-step machine: you feed an input into the first function, then take that result and feed it into the second function. In function notation, applying f after g is written as f(g(x)), which reads “f of g of x”. Composite functions always operate from the inside out – the innermost function is applied first.
当一个函数作用于另一个函数的输出时，就形成了复合函数。这可以想象成一个两步机器：先将输入送进第一个函数，再把结果送进第二个函数。用函数记号表示，先 g 后 f 写作 f(g(x))，读作“f of g of x”。复合函数总是由内向外计算——最内层的函数先作用。
- Example: If f(x) = 2x and g(x) = x + 3, then f(g(x)) = 2(x + 3) = 2x + 6.
- 示例： 若 f(x) = 2x，g(x) = x + 3，则 f(g(x)) = 2(x + 3) = 2x + 6。
The order is crucial: f(g(x)) is usually different from g(f(x)). Always follow the order given in the question.
顺序至关重要：f(g(x)) 通常与 g(f(x)) 不同。务必按照题目给出的顺序运算。
2. Notation: fg(x) and Its Meaning | 记号：fg(x) 及其含义
WJEC exam papers frequently use the compact notation fg(x) to represent the composite function f(g(x)). Do not confuse this with multiplying f and g; it remains a composition where g is applied first, then f. Some texts write f ∘ g, but at GCSE level fg(x) is standard. You may also see expressions like f²(x), which often means ff(x), i.e., applying f twice – check the context carefully.
WJEC 试卷常使用紧凑记号 fg(x) 表示复合函数 f(g(x))。不要把它误解为 f 与 g 相乘；这依然是先算 g 再算 f 的复合。有些教材写作 f ∘ g，但在 GCSE 阶段 fg(x) 最为常见。你还可能见到类似于 f²(x) 的表达式，这通常表示 ff(x)，即两次应用 f——请仔细审题确认。
| Notation | Meaning | Order of Operations |
| fg(x) | f(g(x)) | Apply g, then f |
| gf(x) | g(f(x)) | Apply f, then g |
| ff(x) or f²(x) | f(f(x)) | Apply f twice |
Always write the working step by step to avoid mixing up the direction of substitution.
务必逐步写出代换过程，以免混淆函数作用方向。
3. Evaluating Composite Functions for a Given Value | 对给定数值求复合函数的值
When the question asks, for example, “Find fg(4)”, you must first substitute 4 into the inner function g, find that output, and then use it as the input for f. Using a stepwise method prevents errors.
当题目要求计算例如 fg(4) 时，你必须先将 4 代入内层函数 g，得出结果后再将该结果作为 f 的输入。分步计算可避免出错。
Worked Example: Let f(x) = x² and g(x) = 2x + 1. Find fg(3).
解题示例： 设 f(x) = x²，g(x) = 2x + 1。求 fg(3)。
- Step 1: g(3) = 2(3) + 1 = 7.
- Step 2: f(7) = 7² = 49.
- Therefore, fg(3) = 49.
- 步骤一：g(3) = 2(3) + 1 = 7。
- 步骤二：f(7) = 7² = 49。
- 因此 fg(3) = 49。
Always double-check that you are using the correct function order. If the question gives gf(3), you would do f(3) first, then g.
务必确认你使用的是正确的函数顺序。如果题目给出 gf(3)，则应先计算 f(3)，再计算 g。
4. Forming Composite Functions Algebraically | 用代数方法构建复合函数
Instead of a specific number, you may need to find an algebraic expression for a composite function, such as fg(x). This involves replacing the variable in the outer function with the entire expression of the inner function, then simplifying. Write clear substitution lines to gain full marks in WJEC exams.
除了具体数值，你可能需要求出复合函数的代数表达式，例如 fg(x)。这需要将外层函数中的变量整体替换为内层函数的表达式，然后化简。WJEC 考试中，清晰的代换步骤能帮你拿到满分。
Example: f(x) = 3x − 2, g(x) = x² + 5. Find fg(x) and gf(x).
示例： f(x) = 3x − 2，g(x) = x² + 5。求 fg(x) 和 gf(x)。
- fg(x) = f(g(x)) = f(x² + 5) = 3(x² + 5) − 2 = 3x² + 15 − 2 = 3x² + 13.
- gf(x) = g(f(x)) = g(3x − 2) = (3x − 2)² + 5 = 9x² − 12x + 4 + 5 = 9x² − 12x + 9.
Notice how different the two results are. This emphasizes why careful bracket use in algebra is vital.
注意这两个结果差异明显。这也突出了代数运算中正确使用括号的重要性。
5. Inverse Functions and Their Link to Composition | 反函数及其与复合的联系
nctions. The composition of a function with its inverse brings you back to the starting value: f f⁻¹(x) = f⁻¹f(x) = x. This property often appears in WJEC exam questions that test your understanding of function operations.
反函数用于逆转原函数的效果，记作 f⁻¹(x)。只有一一对应的函数才存在函数意义上的反函数。一个函数与其反函数的复合将返回到初始值：f f⁻¹(x) = f⁻¹f(x) = x。这一性质常出现在 WJEC 考题中，用以考查你对函数运算的理解。
⁻¹(x) = (x − 3)/2. You can check by composing: f(f⁻¹(x)) = 2((x−3)/2) + 3 = x − 3 + 3 = x.
求反函数时，将 y = f(x) 中的 x 与 y 互换，再解出 y。例如 f(x) = 2x + 3，写为 y = 2x + 3，换为 x = 2y + 3，解得 y = (x − 3)/2。因此 f⁻¹(x) = (x − 3)/2。你可以通过复合来检验：f(f⁻¹(x)) = 2((x−3)/2) + 3 = x − 3 + 3 = x。
6. Combining Composite and Inverse Functions | 复合函数与反函数的综合运用
omposite as a single function and work systematically. For fg(x) = a, you can either find the composite expression first and then solve, or peel off layers by using inverse functions – but the algebraic simplification route is safer at GCSE.
WJEC 高级卷常要求解如 fg(x) = a 的方程，或求复合函数的反函数。关键策略是将复合结果视为一个单一函数，再系统求解。对于 fg(x) = a，你可以先求出复合函数表达式再解方程，也可以通过逆运算逐层剥开——但在 GCSE 阶段，代数化简的方法更为稳妥。
Example: f(x) = √x, g(x) = 4x − 1. Solve fg(x) = 3.
示例： f(x) = √x, g(x) = 4x − 1。解 fg(x) = 3。
- Form fg(x) = √4x − 1. Set √4x − 1 = 3.
- Square both sides: 4x − 1 = 9.
- 4x = 10, x = 2.5. Check: g(2.5) = 9, f(9) = √9 = 3. Correct.
- 构建 fg(x) = √4x − 1。令 √4x − 1 = 3。
- 两边平方：4x − 1 = 9。
- 4x = 10，x = 2.5。检验：g(2.5) = 9，f(9) = √9 = 3。正确。
use the rule that (fg)⁻¹(x) = g⁻¹f⁻¹(x). Derive each inverse separately and then compose.
求复合函数的反函数如 (fg)⁻¹(x) 时，可用法则 (fg)⁻¹(x) = g⁻¹f⁻¹(x)。先分别求出各个反函数，再进行复合。
7. Domain and Range Considerations | 定义域与值域的考量
heck for restrictions like square roots (input must be non-negative) or denominators (cannot be zero). For instance, if f(x) = √x and g(x) = x − 5, then fg(x) = √x − 5 requires x − 5 ≥ 0, so x ≥ 5.
复合函数只有在内部函数的输出属于外部函数定义域时才存在。在 GCSE WJEC 中，你主要需要检查类似平方根（输入必须非负）或分母（不能为零）的限制。例如，若 f(x) = √x，g(x) = x − 5，则 fg(x) = √x − 5 要求 x − 5 ≥ 0，即 x ≥ 5。
Sometimes questions will ask for the range of a composite function. You can deduce it by considering the possible outputs of the inner function and how the outer function transforms them. Practising with simple linear and quadratic functions builds confidence for any graph-related domain questions.
有时题目会要求给出复合函数的值域。你可以通过考察内部函数可能的输出以及外部函数如何变换这些输出来推断。通过练习简单的线性与二次函数，你能从容应对任何与图像相关的定义域问题。
8. Common Mistakes and How to Avoid Them | 常见错误及如何避免
ng parentheses when substituting an expression, misinterpreting f²(x) as f(x) × f(x) when it actually means ff(x), and failing to check that the output of the inner function is a valid input for the outer function. Always write the composition out fully before simplifying.
不少考生因复合函数中的简单失误而丢分。最常见的错误包括：颠倒了复合顺序；代入表达式时漏掉括号；错误地把 f²(x) 理解为 f(x) × f(x)，而它实际表示 ff(x)；以及没有检查内部函数的输出是否为外部函数的有效输入。务必在化简前完整写出复合过程。
Pitfall example: is incorrect. The correct composition is f(g(x)) = 1/(x − 3). The brackets are essential.
易错示例： 若 f(x) = 1/x，g(x) = x − 3，写成 fg(x) = 1/x − 3 是错误的。正确的复合为 f(g(x)) = 1/(x − 3)。括号至关重要。
9. Exam-Style Question Practice | 考试型题目演练
Let's work through a typical WJEC higher-tier problem together. The question reads: “ and g(x) = x + 4, find (a) fg(−2), (b) the expression for gf(x), and (c) solve fg(x) = 18.” Set out your solution clearly.
我们一起练习一道典型的 WJEC 高级卷题目。题目为：“已知 f(x) = 2x²，g(x) = x + 4，求 (a) fg(−2)，(b) gf(x) 的表达式，以及 (c) 解方程 fg(x) = 18。”请清晰展示你的解题步骤。
(a) g(−2) = 2, f(2) = 2(2)² = 8. So fg(−2) = 8.
(a) g(−2) = 2，f(2) = 2(2)² = 8。因此 fg(−2) = 8。
(b) gf(x) = g(2x²) = 2x² + 4.
(b) gf(x) = g(2x²) = 2x² + 4。
= 18. Divide by 2: (x + 4)² = 9. Square root both sides: x + 4 = ±3. So x = −1 or x = −7.
(c) fg(x) = 2(x + 4)² = 18。两边除以 2：(x + 4)² = 9。开平方：x + 4 = ±3。得 x = −1 或 x = −7。
Check both solutions back in the original equation to ensure they satisfy the domain and the composite expression.
将两个解代回原方程，确保它们满足定义域及复合表达式。
10. Tips for Success in WJEC Composite Function Questions | WJEC 复合函数题高分技巧
Start by highlighting the order of composition in the question – underline or circle fg(x) and note the direction. Always use substitution lines and check with a numerical trial if time permits. Familiarity with function notation, including the use of f⁻¹(x), will speed up your working. Practise past WJEC papers to get used to how markers expect you to lay out solutions.
答题时先标出题目中的复合顺序——在 fg(x) 下方划线或画圈，并注明方向。始终写出代换过程，若时间允许可用具体数值快速验证。熟悉函数记号，包括 f⁻¹(x) 的运用，能提高你的解题速度。多做 WJEC 历年真题，了解阅卷人期望的解答格式。
ommon errors like sign mistakes. Finally, if the question gives a domain condition, incorporate it into your final answer – for instance, x ≥ 5 for square root contexts.
化简复合函数表达式时，尤其是涉及二次或分式的部分，要仔细展开括号，并警惕符号错误这类常见问题。最后，若题目给出了定义域限制，务必在最终答案中体现——例如在平方根场合加上 x ≥ 5。
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