📚 Composite Functions | 复合函数
Composite functions appear throughout the Edexcel A-Level Mathematics specification, especially in Pure Mathematics. They are used to model successive processes, to understand transformations of graphs, and to explain the relationship between a function and its inverse. A common exam question asks you to form fg(x) or gf(x) from two given functions, to state the domain and range, or to solve an equation involving a composite function. This article builds the topic step by step, using the notation and style expected by Edexcel.
复合函数贯穿 Edexcel A-Level 数学考纲，尤其在纯数学部分。它们用于描述连续过程、理解图像变换，并解释函数与其反函数之间的关系。常见的考试题型包括：根据两个给定函数求 fg(x) 或 gf(x)，写出定义域和值域，或解含有复合函数的方程。本文按照 Edexcel 要求的记号和风格，由浅入深地构建这一主题。
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. If you have two functions f and g, then applying g to x first, and then applying f to the result, produces the composite function written as fg(x). Its value is f(g(x)).
复合函数是将一个函数作用在另一个函数的输出上而得到的函数。如果已知两个函数 f 和 g，先把 g 作用在 x 上，再把 f 作用在所得结果上，就得到复合函数 fg(x)，其值为 f(g(x))。
fg(x) = f(g(x))
For example, if f(x)=x+3 and g(x)=2x, then fg(x)=f(2x)=2x+3. This is not the same as multiplying the two functions; it is one function inside another.
例如，若 f(x)=x+3 且 g(x)=2x，则 fg(x)=f(2x)=2x+3。这不是两个函数相乘，而是一个函数套在另一个函数内部。
2. Notation: fg(x) and f∘g | 记号：fg(x) 与 f∘g
Edexcel usually writes the composite as fg(x), without the small circle symbol. Read fg(x) as ‘f of g of x’ or ‘f after g’. The notation f∘g is equivalent, but you should be comfortable with fg(x) because it is the standard form on Edexcel papers.
Edexcel 通常将复合函数写作 fg(x)，省略小圆圈符号。fg(x) 读作 ‘f of g of x’ 或 ‘f after g’。记号 f∘g 与之等价，但应熟悉 fg(x)，因为这是 Edexcel 试卷的标准写法。
The order in fg(x) can feel unnatural at first: although f is written first, g acts first. Think of fg as ‘g then f’.
fg(x) 中的顺序一开始可能会觉得别扭：虽然 f 写在前面，但 g 先作用。可以把 fg 理解为 ‘先 g 后 f’。
3. Order Matters: fg(x) vs gf(x) | 顺序很重要：fg(x) 与 gf(x)
Composite functions are not generally commutative. In other words, fg(x) and gf(x) usually produce different expressions.
复合函数一般不满足交换律。也就是说，fg(x) 与 gf(x) 通常会得到不同的表达式。
For example, take f(x)=2x+1 and g(x)=x². Then:
例如，取 f(x)=2x+1 和 g(x)=x²，则：
fg(x) = f(x²) = 2x² + 1
gf(x) = g(2x+1) = (2x+1)² = 4x² + 4x + 1
Always check which function is on the inside. In fg(x), g is inside; in gf(x), f is inside.
一定要判断哪个函数在内部。在 fg(x) 中，g 在内部；在 gf(x) 中，f 在内部。
4. Building Composite Functions Algebraically | 用代数方法构造复合函数
To find fg(x), replace every x in f(x) with the entire expression g(x). Use brackets if necessary, then simplify.
求 fg(x) 时，把 f(x) 中的每个 x 替换为整个表达式 g(x)。必要时使用括号，然后化简。
For example, if f(x)=3x-2 and g(x)=x²+1, then:
例如，若 f(x)=3x-2 且 g(x)=x²+1，则：
fg(x) = 3(x² + 1) – 2 = 3x² + 3 – 2 = 3x² + 1
For rational or radical functions, keep the replacement clear. For example, f(x)=1/x and g(x)=x-2 give fg(x)=1/(x-2), not 1/x-2 unless clearly written.
对于有理函数或根式函数，替换要清晰。例如 f(x)=1/x 且 g(x)=x-2，则 fg(x)=1/(x-2)，除非明确写出，否则不要写成 1/x-2。
5. Domain and Range of Composite Functions | 复合函数的定义域与值域
For fg(x) to be defined at a particular x, two conditions must hold: x must belong to the domain of g, and g(x) must belong to the domain of f.
要使 fg(x) 在某一个 x 处有定义，必须满足两个条件：x 必须属于 g 的定义域，且 g(x) 必须属于 f 的定义域。
The domain of fg is therefore the set of x-values in the domain of g whose images under g lie in the domain of f. The range of fg is the set of values f(g(x)) actually takes, and it is a subset of the range of f.
因此 fg 的定义域是 g 的定义域中那些在 g 作用后落入 f 定义域的 x 值的集合。fg 的值域是 f(g(x)) 实际取得的所有值的集合，它是 f 值域的子集。
For example, if f(x)=√x and g(x)=x-3, then fg(x)=√(x-3). We need x-3 ≥ 0, so x ≥ 3. The domain of fg is [3, ∞).
例如，若 f(x)=√x 且 g(x)=x-3，则 fg(x)=√(x-3)。我们需要 x-3 ≥ 0，因此 x ≥ 3。fg 的定义域为 [3, ∞)。
fg(x) = √(x-3), domain: x ≥ 3
6. Existence and Restrictions | 存在性与限制条件
Some functions cannot be composed, or can only be composed after restricting a domain. If the range of g has no values in common with the domain of f, then fg(x) is undefined for every x in the domain of g.
有些函数无法复合，或只有限制定义域后才能复合。如果 g 的值域与 f 的定义域没有公共值，那么对于 g 定义域中的每个 x，fg(x) 都无定义。
It is useful to compare the range of the inner function with the domain of the outer function. For example, f(x)=ln x has domain x>0. If g(x)=-x²-1, then g(x) is always negative, so fg(x)=ln(-x²-1) is undefined for all real x.
比较内层函数的值域与外层函数的定义域很有帮助。例如 f(x)=ln x 的定义域为 x>0。如果 g(x)=-x²-1，则 g(x) 恒为负，因此 fg(x)=ln(-x²-1) 对所有实数 x 都无定义。
7. Composite Functions and Inverse Functions | 复合函数与反函数
If g is the inverse of f, then fg(x)=x for every x in the domain of g, and gf(x)=x for every x in the domain of f. This is the defining property of inverse functions.
如果 g 是 f 的反函数，那么对于 g 定义域中的每个 x，fg(x)=x；对于 f 定义域中的每个 x，gf(x)=x。这就是反函数的定义性质。
For example, f(x)=2x+3 has inverse f⁻¹(x)=(x-3)/2. Then f f⁻¹(x)=x and f⁻¹
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