📚 Composite Functions | 复合函数
In IB Mathematics, composite functions provide a powerful way to link two or more functions together, creating new functions where one function's output becomes another function's input. This concept is central to understanding function transformations, inverses, and advanced calculus topics. Mastering composite functions requires careful attention to notation, domain restrictions, and the order of composition.
在IB数学中，复合函数提供了一种强有力的方法，将两个或多个函数连接起来，创建一个新函数，其中一个函数的输出成为另一个函数的输入。这一概念对于理解函数变换、逆函数以及高等微积分主题至关重要。掌握复合函数需要注意表示法、定义域限制以及复合顺序。
1. Definition and Basic Concept | 定义与基本概念
A composite function is formed when the output of one function becomes the input of another function. Given two functions f and g, the composite function f composed with g, denoted as f ∘ g, is defined by (f ∘ g)(x) = f(g(x)). In this operation, we first apply g to x, and then apply f to the result.
当将一个函数的输出作为另一个函数的输入时，就形成了复合函数。给定两个函数 f 和 g，f 与 g 的复合函数记作 f ∘ g，定义为 (f ∘ g)(x) = f(g(x))。在这个运算中，我们先对 x 应用 g，然后再对结果应用 f。
We can think of a composite function as a 'function machine' with two stages: x → g(x) → f(g(x)). The order of the functions matters, as we will see later.
我们可以将复合函数想象成一个有两个阶段的“函数机器”：x → g(x) → f(g(x))。函数的顺序很重要，我们稍后会看到。
2. Notation: f ∘ g and f(g(x)) | 表示法：f ∘ g 与 f(g(x))
There are two common ways to represent composite functions in IB mathematics. The compact notation f ∘ g reads as 'f composed with g', and (f ∘ alternative form, f(g(x)), directly shows the substitution. Do not confuse the open circle ∘ with multiplication.
IB数学中有两种常见的表示复合函数的方式。紧凑记法 f ∘ g 读作“f 与 g 的复合”，并且 (f ∘ g)(x) = f(g(x))。另一种形式 f(g(x)) 直接显示了代入过程。不要将空心圆点 ∘ 与乘法混淆。
(f ∘ g)(x) = f(g(x))
For example, if f(x) = 2x + 1 and g(x) = x², then (f ∘ g)(x) = f(g(x)) = f(x²) = 2x² + 1.
例如，若 f(x) = 2x + 1，g(x) = x²，则 (f ∘ g)(x) = f(g(x)) = f(x²) = 2x² + 1。
The ∘ symbol reminds us that g acts first, then f. This is sometimes counterintuitive because in 'f ∘ g' the g appears second but is applied first.
∘ 符号提醒我们先执行 g，然后是 f。这有时会与直觉相悖，因为在“f ∘ g”中 g 出现在第二位，但却首先被应用。
3. Domain: The Real Challenge | 定义域：真正的挑战
The domain of a composite function f(g(x)) consists of all x in the domain of g such that g(x) lies in the domain of f. In other words, the input x must first be valid for g, and then the output g(x) must be valid as an input for f.
复合函数 f(g(x)) 的定义域由所有属于 g 的定义域且满足 g(x) 落在 f 的定义域内的 x 组成。换句话说，输入的 x 首先必须在 g 中有效，然后输出 g(x) 必须作为 f 的输入有效。
If either condition fails, the composite function is undefined at that x. This two-step filtering is the most important—and most often examined—aspect of composite functions.
如果任一条件不满足，复合函数在该 x 处无定义。这个两步筛选是复合函数中最重要——也是最常被考查——的方面。
Formally: Dom(f ∘ g) = { x ∈ Dom(g) | g(x) ∈ Dom(f) }.
正式地：Dom(f ∘ g) = { x ∈ Dom(g) | g(x) ∈ Dom(f) }。
Always consider the domain of the inner function first, then impose any restrictions from the outer function on the output of the inner function. Forgetting to check the existence of g(x) in Dom(f) is a common error.
始终先考虑内层函数的定义域，然后根据外层函数的限制对内层函数的输出施加任何约束。忘记检查 g(x) 是否在 Dom(f) 中是常见的错误。
4. How to Find the Domain of a Composite Function | 如何求复合函数的定义域
To determine the domain of (f ∘ g)(x) step by step:
逐步求 (f ∘ g)(x) 的定义域：
- Step 1: Find the domain of the inner function g, say Dg.
步骤1：求出内层函数 g 的定义域，记作 Dg。 - Step 2: Compute the expression g(x) and identify the values for which g(x) belongs to the domain of f.
步骤2：计算表达式 g(x)，并找出使 g(x) 属于 f 的定义域的那些 x 值。 - Step 3: Take the intersection of the conditions from Step 1 and Step 2.
步骤3：取步骤1和步骤2中条件的交集。
Example: Let f(x) = √x (domain: x ≥ 0) and g(x) = x − 4 (domain: all real numbers). Then (f ∘ g)(x) = √x − 4. For f, the input must be ≥ 0, so x − 4 ≥ 0 ⇒ x ≥ 4. The domain
Published by TutorHao | IB 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