Composite Functions | 复合函数 考点精讲

📚 Composite Functions | 复合函数 考点精讲

In GCSE OCR Mathematics, higher-tier students must be confident with composite functions, which involve applying one function to the output of another. Despite the occasional misuse of the term ‘complex functions’ (复变函数), the correct topic is composite functions, usually denoted by fg(x) or f∘g(x). Mastering this concept is vital for algebra, graph transformations, and solving equations.

在GCSE OCR数学中，高阶学生必须熟练掌握复合函数，即把一个函数的输出作为另一个函数的输入。尽管有时被误称为“复变函数”，正确的术语是复合函数，通常记为 fg(x) 或 f∘g(x)。掌握这一概念对于代数、图像变换和解方程至关重要。

1. What Are Composite Functions? | 什么是复合函数？

A composite function is created when one function is applied immediately after another. Given two functions f and g, the composite function ‘f of g of x’ means you first apply g to x, then apply f to the result. Mathematically, (f ∘ g)(x) = f(g(x)).

复合函数是将一个函数紧接着另一个函数作用而得到的新函数。给定两个函数 f 和 g，复合函数“先 g 后 f 作用于 x”意味着先对 x 作用 g，再对结果作用 f。数学上表示为 (f ∘ g)(x) = f(g(x))。

For example, if f(x) = 2x and g(x) = x + 3, then f(g(x)) = 2(x + 3) = 2x + 6. The order matters: g(f(x)) = (2x) + 3 = 2x + 3, which is different.

例如，若 f(x) = 2x，g(x) = x + 3，则 f(g(x)) = 2(x + 3) = 2x + 6。顺序至关重要：g(f(x)) = (2x) + 3 = 2x + 3，两者不同。

2. Notation: fg(x) and f∘g | 符号表示：fg(x) 与 f∘g

OCR exam papers commonly use fg(x) to represent f(g(x)). Be careful not to interpret it as multiplication. The notation f∘g(x) (read ‘f circle g’) is also used. Both mean ‘apply g first, then f’. When you see f²(x), that usually means f(f(x)) – a composite of f with itself.

OCR考试中常使用 fg(x) 表示 f(g(x))，切勿误以为是乘法。符号 f∘g(x)（读作“f 圈 g”）也时有出现，两者均表示“先作用 g，再作用 f”。若见到 f²(x)，通常指 f(f(x))，即 f 与自身的复合。

Always check the context: if f and g are defined, fg(x) must be composite, not product, unless stated otherwise. In function notation, the brackets clarify the order: f(g(x)).

务必结合语境：若已定义 f 和 g，fg(x) 必为复合而非乘积，除非题目另有说明。在函数记号中，括号明确了运算顺序：f(g(x))。

3. How to Form a Composite Function | 如何构造复合函数

To construct f(g(x)), take the expression for g(x) and substitute it wherever you see ‘x’ in f(x). Start from the innermost function and work outward. Write down each step clearly to avoid errors.

构造 f(g(x)) 时，将 g(x) 的表达式代入 f(x) 中所有出现 x 的位置。从最内层函数开始，逐步向外。清晰写出每一步，避免出错。

For example, f(x) = x², g(x) = 3x − 1. Then f(g(x)) = (3x − 1)² = 9x² − 6x + 1. For g(f(x)) = 3(x²) − 1 = 3x² − 1.

例如，设 f(x) = x²，g(x) = 3x − 1。则 f(g(x)) = (3x − 1)² = 9x² − 6x + 1。而 g(f(x)) = 3(x²) − 1 = 3x² − 1。

4. Evaluating Composite Functions Numerically | 数值计算复合函数

Given specific numerical inputs, you can either find the composite algebraic expression first and then substitute, or work step-by-step. Step-by-step: to find fg(2), compute g(2) first, then take that result as the input for f.

给定具体数值时，可以先求出复合后的代数表达式再代入，也可以分步计算。分步法：求 fg(2)，先计算 g(2)，再将结果作为 f 的输入。

Example: f(x) = √x, g(x) = 2x + 3. fg(4) = f(g(4)). g(4) = 2×4 + 3 = 11, then f(11) = √11. So fg(4) = √11.

例：f(x) = √x，g(x) = 2x + 3。fg(4) = f(g(4))。g(4) = 2×4 + 3 = 11，f(11) = √11。故 fg(4) = √11。

Use this method when the functions are complicated or defined by graphs. It minimises algebraic manipulation errors.

当函数较复杂或由图像定义时，可用此方法，它能最大程度减少代数运算错误。

5. Composite Functions with Algebraic Expressions | 代数表达式下的复合函数

When forming composites of algebraic functions, carefully expand and simplify. Watch for brackets and negative signs. For rational functions, combine fractions correctly. Always aim to write the final answer in its simplest form, as required by OCR mark schemes.

对代数函数进行复合时，要小心展开和化简。注意括号和负号。对于有理函数，要正确合并分式。按照OCR评分标准，最终答案应写成最简形式。

Example: f(x) = 2x + a, g(x) = x/2 − b. Find fg(x). fg(x) = 2(x/2 − b) + a = x − 2b + a. Simple.

例：f(x) = 2x + a，g(x) = x/2 − b。求 fg(x)。fg(x) = 2(x/2 − b) + a = x − 2b + a。很简单。Published by TutorHao | GCSE Mathematics Revision Series | aleveler.com

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