📚 Composite Functions | 复合函数
In A-Level Mathematics, composite functions allow us to combine two or more functions to create a new function. Understanding how to form, evaluate, and manipulate composite functions is essential for success in Edexcel Pure Mathematics, as they appear in topics such as algebra, calculus, and transformations.
在A-Level数学中，复合函数使我们能够将两个或多个函数组合起来，构成一个新函数。理解如何构造、计算和操作复合函数对于在爱德思纯数学考试中取得成功至关重要，因为它们出现在代数、微积分和变换等主题中。
1. What Are Composite Functions? | 什么是复合函数？
ion f(g(x)) means we first apply g to x, then apply f to the result. This is often read as 'f of g of x'. The notation f ∘ g denotes the composite function, so (f ∘ g)(x) = f(g(x)).
复合函数是指一个函数的输出成为另一个函数的输入。如果我们有两个函数 f 和 g，那么复合函数 f(g(x)) 表示先将 g 作用于 x，再将 f 作用于得到的结果。这通常读作“f 圈 g 作用于 x”。符号 f ∘ g 表示复合函数，因此 (f ∘ g)(x) = f(g(x))。
It is important to note that the order matters: f(g(x)) is generally not the same as g(f(x)). The function on the right, g, is applied first, then f. For the composite f(g(x)) to be defined, the range of g must overlap with the domain of f.
重要的是注意顺序：f(g(x)) 通常不等于 g(f(x))。右边的函数 g 首先被应用，然后是 f。要使复合 f(g(x)) 有定义，g 的值域必须与 f 的定义域有交集。
2. Notation and Terminology | 符号与术语
The standard notation for a composite function is (f ∘ ten without the circle, simply as fg(x), but this can be ambiguous with multiplication, so in Edexcel exams you will see either f(g(x)) or f ∘ g(x). The circle ∘ means 'composed with'. Remember that f ∘ g means apply g first, then f.
复合函数的标准符号是 (f ∘ g)(x) = f(g(x))。有时没有圆圈，直接写成 fg(x)，但这容易与乘法混淆，因此在爱德思考试中你会看到 f(g(x)) 或 f ∘ g(x)。圆圈 ∘ 表示“与…复合”。记住，f ∘ g 表示先应用 g，再应用 f。
(f ∘ g)(x) = f(g(x))
When we talk about the domain of the composite function, we must consider the domain of the inner function and any additional restrictions imposed by the outer function. The composite function f(g(x)) exists only for x in the domain of g such that g(x) is in the domain of f.
当我们谈论复合函数的定义域时，必须考虑内层函数的定义域以及外层函数施加的任何额外限制。复合函数 f(g(x)) 仅对于属于 g 的定义域且 g(x) 在 f 的定义域内的那些 x 值存在。
3. Forming Composite Functions Step by Step | 逐步构造复合函数
To form f(g(x)) given f(x) and g(x):
给定 f(x) 和 g(x)，构造 f(g(x)) 的步骤：
Write g(x) explicitly, e.g., g(x) = 2x + 3.
明确写出 g(x)，例如 g(x) = 2x + 3。
g(x) into f(x) wherever you see x in f. If f(x) = x², then f(g(x)) = (g(x))² = (2x + 3)².
将 f(x) 中所有的 x 替换为 g(x)。若 f(x) = x²，则 f(g(x)) = (g(x))² = (2x + 3)²。
Simplify the resulting expression.
化简所得表达式。
uting. For example, if f(x) = 1/x and g(x) = x - 1, then f(g(x)) = 1/(x - 1), not 1/x - 1.
代入时务必注意括号。例如，若 f(x) = 1/x，g(x) = x - 1，则 f(g(x)) = 1/(x - 1)，而不是 1/x - 1。
4. Domain and Range of Composite Functions | 复合函数的定义域与值域
The domain of the composite function f(g(x)) is the set of all x in the domain of g for which g(x) is in the domain of f. This often requires solving inequalities to ensure the inner output is valid for the outer function.
复合函数 f(g(x)) 的定义域是 g 的定义域中所有使得 g(x) 在 f 的定义域中的 x 的集合。这通常需要解不等式以确保内层输出对外层函数有效。
For example, let f(x) = √x (domain x ≥ 0) and g(x) = x - 4 (domain all real numbers). Then f(g(x)) = √x - 4 x - 4 ≥ 0, so x ≥ 4. Thus the domain of f(g(x)) is [4, ∞). Here g(x) = x - 4 must be ≥ 0.
例如，设 f(x) = √x（定义域 x ≥ 0），g(x) = x - 4（定义域全体实数）。那么 f(g(x)) = √x - 4。定义域要求 x - 4 ≥ 0，即 x ≥ 4。因此 f(g(x)) 的定义域为 [4, ∞)。这里 g(x) = x - 4 必须 ≥ 0。
The range of the composite function can often be found by considering the range of the inner function and then applying the outer function, or by analysing the simplified expression.
复合函数的值域通常可通过考虑内层函数的值域再应用外层函数，或通过分析化简后的表达式求得。
5. Evaluating Composite Functions at a Point | 计算复合函数在某点的值
thod: compute b = g(a), then compute f(b). This is often quicker for numerical evaluation.
要计算给定数值 a 处的 f(g(a))，可以从内向外计算，或者先求出复合函数表达式。从内向外法：计算 b = g(a)，然后计算 f(b)。对于数值计算，这通常更快捷。
(x) = 2x + 5, and a = 2, then g(2) = 4 + 1 = 5, then f(5) = 2×5 + 5 = 15, so f(g(2)) = 15.
例如，若 g(x) = x² + 1，f(x) = 2x + 5，a = 2，则 g(2) = 4 + 1 = 5，然后 f(5) = 2×5 + 5 = 15，因此 f(g(2)) = 15。
Ensure you follow the correct order. Mixing up the order, such as g(f(x)) instead of f(g(x)), will give a different result.
一定要遵循正确的顺序
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图