📚 Composite Functions | 复合函数考点精讲
Composite functions are built by applying one function to the output of another. They appear frequently in GCSE Mathematics and are essential for understanding function notation, domain restrictions, and algebraic manipulation. Mastering composite functions will sharpen your skills in substitution and help you tackle higher-tier exam questions with confidence.
复合函数是将一个函数作用于另一个函数的输出所构成的函数。它们在 GCSE 数学中经常出现，对于理解函数符号、定义域限制和代数操作至关重要。掌握复合函数可以提升你的代入技巧，并帮助你自信地应对高等级考试题目。
1. What Is a Composite Function? | 什么是复合函数？
A composite function is formed when the output of one function becomes the input of another. If you have two functions f and g, the composite function f of g of x is written as f(g(x)). This means you first apply g to x, then apply f to the result. The process is sometimes described as a “function machine” with two stages.
复合函数是指将一个函数的输出作为另一个函数的输入而形成的函数。如果有两个函数 f 和 g，复合函数 f 于 g 于 x 记作 f(g(x))。这意味着先将 g 作用于 x，再将 f 作用于得到的结果。这个过程有时被描述为带有两个阶段的“函数机器”。
For example, let f(x) = 3x + 1 and g(x) = x². Then f(g(x)) = f(x²) = 3(x²) + 1 = 3x² + 1. Here, g turns x into x², and f then multiplies by 3 and adds 1.
例如，设 f(x) = 3x + 1，g(x) = x²。那么 f(g(x)) = f(x²) = 3(x²) + 1 = 3x² + 1。此处 g 将 x 变成 x²，然后 f 乘 3 再加 1。
2. Notation and Reading fg(x) | 符号表示与 fg(x) 的读法
GCSE exam papers often use the notation fg(x) to mean f(g(x)). Note that fg(x) does not mean f(x) times g(x); it means apply g first, then f. The order is right to left: start with the inner function closest to x.
GCSE 试卷常使用 fg(x) 表示 f(g(x))。注意 fg(x) 并不意味着 f(x) 乘以 g(x)；它表示先应用 g，再应用 f。顺序是从右到左：从最靠近 x 的内部函数开始。
Some students confuse fg(x) with gf(x). They are generally different. For f(x) = 2x and g(x) = x + 3, fg(x) = f(x+3) = 2(x+3) = 2x+6, while gf(x) = g(2x) = 2x+3. Always check the order carefully.
一些学生会混淆 fg(x) 和 gf(x)。它们通常不相同。对于 f(x) = 2x 和 g(x) = x + 3，fg(x) = f(x+3) = 2(x+3) = 2x+6，而 gf(x) = g(2x) = 2x+3。务必仔细检查顺序。
3. Evaluating Composite Functions Step by Step | 逐步求复合函数的值
To evaluate a composite function at a specific number, work from the inside out. Suppose f(x) = 5x − 2 and g(x) = x² + 1. Find fg(3). First calculate g(3) = 3² + 1 = 10. Then substitute into f: f(10) = 5(10) − 2 = 48. So fg(3) = 48.
要求复合函数在具体数值处的值，要从内向外计算。假设 f(x) = 5x − 2，g(x) = x² + 1。求 fg(3)。首先计算 g(3) = 3² + 1 = 10。然后代入 f：f(10) = 5(10) − 2 = 48。所以 fg(3) = 48。
You can also form the algebraic expression first. For fg(x) = f(x²+1) = 5(x²+1) − 2 = 5x² + 5 − 2 = 5x² + 3. Then substitute x = 3: 5(9) + 3 = 45 + 3 = 48. Both methods give the same answer.
你也可以先求出代数表达式。fg(x) = f(x²+1) = 5(x²+1) − 2 = 5x² + 5 − 2 = 5x² + 3。然后代入 x = 3：5(9) + 3 = 45 + 3 = 48。两种方法得到相同答案。
4. Domain and Range of Composite Functions | 复合函数的定义域与值域
The domain of a composite function fg(x) is the set of all x such that x is in the domain of g, and g(x) is in the domain of f. This means we must exclude any x-values that make g undefined or lead to an input that f cannot accept.
复合函数 fg(x) 的定义域是使得 x 属于 g 的定义域，且 g(x) 属于 f 的定义域的所有 x 的集合。这意味着我们必须排除那些使 g 无定义，或导致 f 无法接受的输入值的 x。
For instance, if f(x) = 1/(x-2) and g(x) = √x, then for fg(x) = 1/(√x – 2) we need x ≥ 0 (from the square root) and √x ≠ 2, so x ≠ 4. Thus the domain is x ≥ 0, x ≠ 4. Always consider restrictions from both functions.
例如，若 f(x) = 1/(x-2)，g(x) = √x，则对于 fg(x) = 1/(√x – 2)，我们需要 x ≥ 0（来自平方根）且 √x ≠ 2，所以 x ≠ 4。因此定义域为 x ≥ 0 且 x ≠ 4。务必同时考虑两个函数的限制。
5. Composing a Function with Itself | 函数自身的复合
You can form ff(x), which means f(f(x)). This is useful for transformations and iteration. If f(x) = 2x + 3, then ff(x) = f(2x+3) = 2(2x+3) + 3 = 4x + 6 + 3 = 4x + 9. Such expressions often appear in higher-tier questions involving functional equations.
你可以构造 ff(x)，它表示 f(f(x))。这对变换和迭代很有用。如果 f(x) = 2x + 3，那么 ff(x) = f(2x+3) = 2(2x+3) + 3 = 4x + 6 + 3 = 4x + 9。这类表达式常出现在涉及函数方程的高等级题目中。
Similarly, gg(x) or even more complex chains can be built. For example, if g(x) = x² − 1, then ggg(x) means g(g(g(x))). Practice with two or three applications to gain fluency.
类似地，可以构造 gg(x) 甚至更复杂的链。例如，如果 g(x) = x² − 1，那么 ggg(x) 表示 g(g(g(x)))。通过两到三次应用的练习来提升熟练度。
6. Inverse Functions and Composite Functions | 反函数与复合函数
A function f and its inverse f⁻¹ undo each other, so f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This relationship is often tested in combination with composite functions. If you are given fg(x) and need to find g⁻¹, you can form f(g(x)) = h(x) and rearrange.
函数 f 与其反函数 f⁻¹ 互为逆运算，因此 f(f⁻¹(x)) = x 且 f⁻¹(f(x)) = x。这种关系经常与复合函数结合考查。如果已知 fg(x)，需要求 g⁻¹，可以利用 f(g(x)) = h(x) 并整理求解。
For example, if f(x) = 2x + 5 and fg(x) = 6x + 1, find g(x). Since fg(x) = f(g(x)) = 2g(x) + 5 = 6x + 1, we solve 2g(x) = 6x – 4, so g(x) = 3x – 2. Then g⁻¹(x) can be found by switching x and y: y = 3x – 2 becomes x = 3y – 2, so y = (x+2)/3.
例如，如果 f(x) = 2x + 5 且 fg(x) = 6x + 1，求 g(x)。因为 fg(x) = f(g(x)) = 2g(x) + 5 = 6x + 1，我们解出 2g(x) = 6x – 4，所以 g(x) = 3x – 2。然后 g⁻¹(x) 可通过交换 x 与 y 求得：y = 3x – 2 变为 x = 3y – 2，所以 y = (x+2)/3。
7. Solving Equations with Composite Functions | 解含有复合函数的方程
Exam questions may give two functions and an equation like fg(x) = gf(x). You need to compute both sides and solve for x. For f(x) = x + 4 and g(x) = 3x, fg(x) = (3x) + 4 = 3x + 4, gf(x) = 3(x+4) = 3x + 12. Setting them equal gives 3x + 4 = 3x + 12, which has no solution. Be ready for such inconsistent cases.
考试题可能给出两个函数以及形如 fg(x) = gf(x) 的方程。你需要计算两边并求解 x。对于 f(x) = x + 4 和 g(x) = 3x，fg(x) = (3x) + 4 = 3x + 4，gf(x) = 3(x+4) = 3x + 12。令两者相等得到 3x + 4 = 3x + 12，无解。要做好遇到这种矛盾情况的准备。
More commonly, you will solve fg(x) = k, where k is a number. Example: f(x) = 2x − 1, g(x) = x², solve fg(x) = 7. First, fg(x) = 2x² − 1 = 7, so 2x² = 8, x² = 4, x = ±2. Always check domain constraints to confirm solutions are valid.
更常见的是求解 fg(x) = k，其中 k 是一个常数。例如：f(x) = 2x − 1，g(x) = x²，求解 fg(x) = 7。首先，fg(x) = 2x² − 1 = 7，得 2x² = 8，x² = 4，x = ±2。务必检查定义域限制以确认解的有效性。
8. Common Pitfalls and How to Avoid Them | 常见陷阱及其避免方法
Many students mistakenly multiply functions instead of composing them. Remember: fg(x) ≠ f(x) × g(x). Always substitute the inner function into the outer function. Another error is misreading the order; fg(x) means apply g first, not f first.
许多学生错误地将函数相乘而不是复合。请记住：fg(x) ≠ f(x) × g(x)。务必将内层函数代入外层函数。另一个错误是弄错顺序；fg(x) 表示先应用 g，而非先应用 f。
Domain oversights are also common. After finding an expression for fg(x), check for any restrictions from the interior and exterior functions. For example, if g involves a denominator, ensure that denominator is not zero, and also ensure the output does not violate f’s domain.
忽视定义域也比较常见。在求得 fg(x) 的表达式后，要检查内、外层函数的任何限制。例如，若 g 含有分母，要确保分母不为零，也要确保输出不违反 f 的定义域。
9. Tables of Values and Graphs of Composite Functions | 复合函数的数值表与图像
You can construct a table of values for a composite function by first computing the inner function, then the outer. This helps in sketching graphs and understanding transformations. For f(x) = sin x and g(x) = 2x, fg(x) = sin(2x) compresses the graph horizontally by factor 1/2.
你可以通过先计算内层函数，再计算外层函数来构建复合函数的数值表。这有助于绘制图像并理解变换。对于 f(x) = sin x 和 g(x) = 2x，fg(x) = sin(2x) 将图像水平压缩为原来的 1/2。
Using a simple table:
| x | g(x)=2x | f(g(x))=sin(2x) |
|---|---|---|
| 0 | 0 | 0 |
| π/4 | π/2 | 1 |
| π/2 | π | 0 |
Interpreting such tables strengthens the link between algebraic and graphical reasoning.
解读这类表格能加强代数与图形推理之间的联系。
10. Real-World Applications and Modelling | 实际应用与建模
Composite functions model sequences of real-world processes. For example, a discount followed by tax: if a £x item has a 20% discount function d(x)=0.8x, then sales tax t(x)=1.05x, the final price is t(d(x))=1.05(0.8x)=0.84x. This composite reflects both operations in order.
复合函数可以对现实世界中的连续过程进行建模。例如，先打折再加税：如果一件商品价格为 £x，享受八折优惠函数 d(x)=0.8x，然后加上销售税 t(x)=1.05x，最终价格为 t(d(x))=1.05(0.8x)=0.84x。这个复合函数按顺序反映了两种操作。
Understanding composites helps in areas like currency conversion with fees, or converting units where one function changes metres to centimetres and another converts to inches. Gaining fluency in function composition makes these multi-step problems straightforward.
理解复合函数有助于处理如带手续费的货币兑换，或一个函数将米转换为厘米、另一个转换为英寸的问题。熟练掌握函数的复合能让这些多步骤问题变得简单明了。
11. Exam Tips and Tricky Question Types | 考试技巧与难题类型
When given fg(x) = expression and one of the functions, you may need to find the other function by working backwards. If f is unknown but g is known, let f(x)=… and substitute g(x) to form an equation. Matching coefficients often helps.
当已知 fg(x) 的表达式和其中一个函数时，你可能需要通过逆向推导找出另一个函数。如果 f 未知但 g 已知，可设 f(x)=… 然后代入 g(x) 得到方程。匹配系数通常会有所帮助。
Always watch for domain clues: a function like g(x)=√(x-3) demands x≥3, and if f involves division, exclude values making denominators zero. Write down all restrictions before finalising your answer.
始终留意定义域线索：像 g(x)=√(x-3) 这样的函数要求 x≥3，如果 f 包含除法，要排除使分母为零的值。在最终确定答案之前，写下所有限制条件。
12. Summary and Key Points | 总结与核心要点
Composite functions combine two or more functions into a single process. Always work from the innermost function outward. fg(x) means f(g(x)), not multiplication. Check domain restrictions for both functions. Use algebraic substitution to evaluate, build equations, or find unknown functions.
复合函数将两个或多个函数组合成一个过程。始终从最内层函数向外计算。fg(x) 表示 f(g(x))，而不是相乘。检查两个函数的定义域限制。使用代数代入求值、建立方程或找出未知函数。
Practise a wide variety of questions, including those with fractions, squares, and trigonometric functions, to become fully prepared. With clear understanding and careful working, composite functions become a reliable source of marks.
练习各种类型的题目，包括带分数、平方和三角函数的题目，以做好充分准备。有了清晰的理解和细致的计算，复合函数会成为一个可靠的得分点。
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