Composite Functions: Operations and Applications | 复合函数的运算与应用

📚 Composite Functions: Operations and Applications | 复合函数的运算与应用

Composite functions are a key topic in the Edexcel IGCSE Mathematics syllabus. They allow us to combine two or more functions in a single expression, applying one rule and then the next. This article explains how to work with composite functions, including order, notation, domains, and practical applications.

复合函数是 Edexcel IGCSE 数学大纲中的一个重点专题。它让我们能把两个或更多函数结合成一个表达式，先应用一个规则，再应用下一个规则。本文将解释如何运算复合函数，包括顺序、记号、定义域与实际应用。

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 function. For example, if f and g are two functions, then the composite function written as f(g(x)) means: first apply g to x, then apply f to the result.

复合函数是指一个函数的输出变成另一个函数的输入。例如，若 f 和 g 是两个函数，则记作 f(g(x)) 的复合函数表示：先把 g 作用于 x，再把 f 作用于所得结果。

Imagine a machine with two stages. An input x enters the first machine g, and g produces an output g(x). That output then enters the second machine f, which produces the final output f(g(x)).

可以把复合函数想象成一台两段式机器。输入 x 先进入第一台机器 g，g 产生输出 g(x)；这个输出再进入第二台机器 f，最终得到输出 f(g(x))。

In Edexcel IGCSE notation, f(g(x)) is often written as fg(x) or f ∘ g(x). All of these mean exactly the same thing.

在 Edexcel IGCSE 的记号中，f(g(x)) 通常写成 fg(x) 或 f ∘ g(x)。这些写法表示完全相同的含义。

2. Notation and Order | 复合函数的记号与顺序

For two functions f and g, the composite function fg(x) is defined as f(g(x)). The function written closest to x is applied first. Because of this, we evaluate composite functions from right to left.

对两个函数 f 和 g，复合函数 fg(x) 定义为 f(g(x))。离 x 最近的函数最先被应用。因此，我们计算复合函数时要从右往左进行。

fg(x) = f(g(x))

Swapping the order can change the result completely. In general, fg(x) is not equal to gf(x).

交换两个函数的顺序可能使结果完全不同。一般情况下，fg(x) 不等于 gf(x)。

For example, let f(x) = 2x and g(x) = x + 1. Then:

例如，设 f(x) = 2x，g(x) = x + 1。则：

fg(3) = f(g(3)) = f(4) = 8

gf(3) = g(f(3)) = g(6) = 7

This shows why order matters. In most problems you must read the notation carefully before starting.

这说明了顺序为什么重要。在多数题目中，你必须先仔细辨认记号再开始计算。

3. How to Evaluate Composite Functions | 如何计算复合函数

There are two common methods for evaluating a composite function. The first is to work step by step with numbers; the second is to find an algebraic expression first.

计算复合函数有两种常用方法。第一种是逐步代入数值；第二种是先求出代数表达式。

Step-by-step method for fg(2):

逐步求值法计算 fg(2)：

Identify the inner function. In fg(x), the inner function is g(x).
确定内层函数。在 fg(x) 中，内层函数是 g(x)。
Evaluate g(2) first.
先计算 g(2)。
Substitute this value into f.
把这个值代入 f。

Example: f(x) = 3x + 1 and g(x) = x². Find fg(2).

例：f(x) = 3x + 1，g(x) = x²。求 fg(2)。

g(2) = 2² = 4

f(4) = 3 × 4 + 1 = 13

So fg(2) = 13. This method is useful when you only need one value.

所以 fg(2) = 13。当你只需要一个数值时，这种方法很方便。

For more than one value, it is often faster to combine the functions algebraically.

若需要计算多个值，通常先用代数方法合并函数会更快捷。

4. Finding fg(x) for Given Functions | 求给定函数的复合表达式

To find an algebraic expression for fg(x), replace the x in f with the whole expression g(x).

要求 fg(x) 的代数表达式，只需把 f 中的 x 替换成 g(x) 的完整表达式。

Example: f(x) = 3x − 2 and g(x) = x² + 1. Find fg(x).

例：f(x) = 3x − 2，g(x) = x² + 1。求 fg(x)。

fg(x) = f(x² + 1) = 3(x² + 1) − 2

fg(x) = 3x² + 3 − 2 = 3x² + 1

Now find gf(x). This time, replace x in g with f(x).

现在求 gf(x)。这一次，把 g 中的 x 替换成 f(x)。

gf(x) = g(3x − 2) = (3x − 2)² + 1

gf(x) = 9x² − 12x + 4 + 1 = 9x² − 12x + 5

Notice that fg(x) and gf(x) are different expressions. You should always expand brackets carefully, especially squares such as (3x − 2)².

注意 fg(x) 和 gf(x) 是两个不同的表达式。你应当小心展开括号，尤其是 (3x − 2)² 这样的平方。

5. Domain and Range of Composite Functions | 复合函数的定义域与值域

The domain of a function is the set of values that can be put into the function. For a composite function fg(x), the input x must be in the domain of g, and g(x) must be in the domain of f.

函数的定义域是可以输入函数的值集。对复合函数 fg(x)，输入 x 必须在 g 的定义域内，并且 g(x) 必须在 f 的定义域内。

Example: f(x) = √x and g(x) = x − 3. Find fg(x) and its domain.

例：f(x) = √x，g(x) = x − 3。求 fg(x) 及其定义域。

fg(x) = f(x − 3) = √(x − 3)

Because the square root is only defined for non-negative numbers, we require x − 3 ≥ 0, so x ≥ 3. Therefore the domain of fg(x) is x ≥ 3.

因为根号内必须是非负数，所以要求 x − 3 ≥ 0，即 x ≥ 3。因此 fg(x) 的定义域是 x ≥ 3。

The range of fg(x) is the set of possible final outputs. In this example, √(x − 3) can produce every value from 0 upward, so the range is y ≥ 0.

fg(x) 的值域是最终输出值组成的集合。在本例中，√(x − 3) 能从 0 开始取到所有非负值，因此值域为 y ≥ 0。

When functions are linear, the domain is usually all real numbers, but you should still check for restrictions such as square roots or denominators.

当函数是线性函数时，定义域通常为全体实数，但你仍然需要检查根号或分母等限制条件。

6. Composite Functions with Numerical Inputs | 复合函数的数值代入

A table can help organise your work when evaluating several inputs.

在计算多个输入值时，表格可以帮助你整理思路。

Let f(x) = x + 5 and g(x) = 2x. Evaluate fg(x) for x = 0, 1, and 2.

设 f(x) = x + 5，g(x) = 2x。分别求 x = 0、1、2 时的 fg(x)。

x	g(x) = 2x	f(g(x)) = g(x) + 5
0	0	5
1	2	7
2	4	9

Alternatively, first combine the functions:

或者，先合并函数：

fg(x) = f(2x) = 2x + 5

Then fg(0) = 5, fg(1) = 7, and fg(2) = 9. Both approaches give the same answers.

于是 fg(0) = 5，fg(1) = 7，fg(2) = 9。两种方法得到相同的结果。

7. Inverse Functions and Composite Functions | 反函数与复合函数

A function and its inverse are closely connected through composition. If f⁻¹ is the inverse of f, then composing them in either order gives the identity function.

一个函数与其反函数通过复合密切关联。若 f⁻¹ 是 f 的反函数，则按任意顺序复合它们都会得到恒等函数。

f⁻¹f(x) = x

ff⁻¹(x) = x

Example: f(x) = 2x + 3. Find f⁻¹(x) and check f⁻¹f(x).

例：f(x) = 2x + 3。求 f⁻¹(x) 并验证 f⁻¹f(x)。

To find the inverse, write y = 2x + 3, then swap x and y and solve for y.

求反函数时，先写 y = 2x + 3，再交换 x 和 y，并解出 y。

x = 2y + 3

y = (x − 3) ÷ 2

f⁻¹(x) = (x − 3) ÷ 2

Now check the composition:

现在验证复合：

f⁻¹f(x) = f⁻¹(2x + 3) = ((2x + 3) − 3) ÷ 2 = 2x ÷ 2 = x

This confirms that f⁻¹ undoes f. In Edexcel IGCSE questions, you may be asked to find an inverse and then use composition to check your answer.

这验证了 f⁻¹ 能抵消 f 的作用。在 Edexcel IGCSE 题目中，你可能会被要求求反函数并用复合运算检查答案。

8. Self-Composition: ff(x) | 函数的自复合：ff(x)

Sometimes a function is composed with itself. This is called self-composition and is written as ff(x) or f²(x). It means apply f, then apply f again.

有时函数会与自身复合。这称为自复合，记作 ff(x) 或 f²(x)。它表示先应用 f，再应用一次 f。

Example: f(x) = x² + 1. Find ff(x).

例：f(x) = x² + 1。求 ff(x)。

ff(x) = f(x² + 1) = (x² + 1)² + 1

ff(x) = x⁴ + 2x² + 1 + 1 = x⁴ + 2x² + 2

Be careful: f²(x) here does not mean f(x) × f(x) in this context. It means f(f(x)). This is standard notation in IGCSE.

注意：这里的 f²(x) 并不表示 f(x) × f(x)，而是表示 f(f(x))。这是 IGCSE 中的标准记号。

Self-composition appears in problems involving repeated processes, such as applying the same percentage growth twice.

自复合常出现在涉及重复过程的题目中，例如同一百分比增长连续应用两次。

9. Real-World Applications | 复合函数的实际应用

Composite functions are useful in real life whenever one quantity depends on another through more than one stage.

当一个量需要经过多个阶段依赖于另一个量时，复合函数在现实生活中非常有用。

Example: A shop offers a discount of 10 pounds on every item, and then adds 20% tax.

例：一家商店每件商品先优惠 10 英镑，然后加收 20% 的税。

Let d(x) = x − 10 represent the discount applied first, and t(x) = 1.2x represent the tax applied to the discounted price. The overall price is the composite function:

设 d(x) = x − 10 表示先应用优惠，t(x) = 1.2x 表示对折后价格加税。最终价格就是复合函数：

t(d(x)) = t(x − 10) = 1.2(x − 10)

If the original price is 50 pounds, then t(d(50)) = 1.2 × 40 = 48 pounds.

若原价为 50 英镑，则 t(d(50)) = 1.2 × 40 = 48 英镑。

If the tax is applied first and the discount second, the composite is d(t(x)) = 1.2x − 10. This gives a different result: 1.2 × 50 − 10 = 50 pounds.

如果先加税再优惠，复合函数为 d(t(x)) = 1.2x − 10。这会得到不同的结果：1.2 × 50 − 10 = 50 英镑。

This example shows that real-world order matters. In exam questions, always decide which function is applied first by reading the words carefully.

这个例子说明现实中的顺序很重要。在考试题目中，务必通过仔细阅读文字来决定先应用哪个函数。

10. Common Mistakes | 常见错误

Many students lose marks on composite functions because of small but important errors.

许多学生在复合函数题中失分，是因为一些细微但重要的错误。

Applying the wrong function first. In fg(x), apply g first and f second.
先应用了错误的函数。在 fg(x) 中，应先应用 g，再应用 f。
Forgetting to replace every x. If f(x) = 3x + 2, then f(x²) = 3x² + 2, not 3x + 2.
忘记替换每一个 x。若 f(x) = 3x + 2，则 f(x²) = 3x² + 2，而不是 3x + 2。
Expanding squares incorrectly. Remember (x + 1)² = x² + 2x + 1, not x² + 1.
错误展开平方。记住 (x + 1)² = x² + 2x + 1，而不是 x² + 1。
Ignoring domain restrictions when the composite involves roots or fractions.
当复合函数涉及根号或分数时，忽略了定义域限制。

To avoid these mistakes, write out the composite expression clearly before simplifying.

要避免这些错误，请先把复合表达式写清楚，然后再化简。

11. Exam-Style Worked Example | 考试真题详解

Let us work through a typical Edexcel IGCSE style question.

让我们一起完成一道典型的 Edexcel IGCSE 风格题目。

Question: f(x) = 3x + 2 and g(x) = x² − 1.

题目：f(x) = 3x + 2，g(x) = x² − 1。

(a) Find fg(x).

(a) 求 fg(x)。

fg(x) = f(x² − 1) = 3(x² − 1) + 2 = 3x² − 3 + 2 = 3x² − 1

(b) Find gf(2).

(b) 求 gf(2)。

First find f(2):

先求 f(2)：

f(2) = 3 × 2 + 2 = 8

gf(2) = g(8) = 8² − 1 = 64 − 1 = 63

Alternatively, first find the algebraic expression gf(x):

或者先求代数表达式 gf(x)：

gf(x) = g(3x + 2) = (3x + 2)² − 1

gf(x) = 9x² + 12x + 4 − 1 = 9x² + 12x + 3

Then gf(2) = 9 × 4 + 12 × 2 + 3 = 36 + 24 + 3 = 63. Both methods agree.

然后 gf(2) = 9 × 4 + 12 × 2 + 3 = 36 + 24 + 3 = 63。两种方法答案一致。

3x² − 1 = 0

3x² = 1

x² = 1 ÷ 3

x = ±√(1 ÷ 3)

Always check whether your answer form matches the question, such as surd form or decimal form.

始终检查答案形式是否符合题目要求，例如根式形式或小数形式。

12. Summary | 总结

Composite functions combine two functions by applying one after the other. The notation fg(x) means f(g(x)), and the function closest to x is applied first.

复合函数通过按顺序应用两个函数来将它们结合。记号 fg(x) 表示 f(g(x))，离 x 最近的函数最先被应用。

To evaluate a composite function, you can work numerically step by step or build an algebraic expression first. Order matters, so fg(x) and gf(x) are usually different.

计算复合函数时，你可以逐步代入数值，也可以先建立代数表达式。顺序很重要，因此 fg(x) 与 gf(x) 通常不同。

Remember to check domains, expand brackets carefully, and use inverse composition to verify inverse functions. With regular practice, composite functions become a reliable source of marks in the IGCSE exam.

请记得检查定义域、小心展开括号，并利用复合运算验证反函数。通过规律练习，复合函数会成为 IGCSE 考试中一个稳定的得分点。

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Composite Functions: Operations and Applications | 复合函数的运算与应用