IGCSE Additional Mathematics: Functions Exam Essentials | IGCSE附加数学：函数考点精讲

📚 IGCSE Additional Mathematics: Functions Exam Essentials | IGCSE附加数学：函数考点精讲

Functions are one of the most important topics in IGCSE Additional Mathematics. Many higher-level questions are built around the key ideas of domain and range, composite functions and inverse functions. In this article, we will break down every examinable skill step by step, with bilingual explanations to help you master the topic quickly.

函数是IGCSE附加数学中最重要的考点之一。许多高阶题目都围绕定义域与值域、复合函数和反函数等核心概念展开。在本文中，我们将逐一拆解所有考试技能，配合中英双语讲解，帮助你快速掌握这一主题。

1. Function Notation and Basics | 函数符号与基础

A function is a rule that assigns exactly one output to each valid input. We write y = f(x), where x is the independent variable and y is the dependent variable. The key requirement is that each x-value in the domain produces only one y-value.

函数是一种规则，它为每一个有效输入恰好分配一个输出。我们写作 y = f(x)，其中 x 是自变量，y 是因变量。关键在于定义域中的每一个 x 值只能产生一个 y 值。

You should be able to evaluate a function at a number or an expression. For example, if f(x) = 3x + 2, then f(2) = 8 and f(a + 1) = 3(a + 1) + 2 = 3a + 5. Notice that the entire input replaces x.

你应该能够计算函数在某个数或某个表达式上的值。例如，若 f(x) = 3x + 2，则 f(2) = 8，f(a + 1) = 3(a + 1) + 2 = 3a + 5。注意整个输入都会替换 x。

A graph represents a function only if every vertical line crosses the graph at most once. This is called the vertical line test. If a vertical line hits two points, the relation is not a function.

一个图像只有在每条竖直线最多与图像相交一次时，才表示函数。这称为垂线检验。如果一条竖直线与图像相交于两点，则该关系不是函数。

2. Domain and Range | 定义域与值域

The domain is the set of all x-values for which the function is defined. The range is the set of all y-values that the function can output.

定义域是使函数有意义的所有 x 值组成的集合。值域是函数所有可能输出的 y 值组成的集合。

There are two main restrictions in IGCSE Additional Mathematics: denominators cannot be zero, and square roots of negative numbers are not allowed. For example, f(x) = 1/(x – 2) has domain x ≠ 2. For f(x) = √(x – 3), we need x ≥ 3.

在IGCSE附加数学中主要有两种限制：分母不能为零，以及负数不能开平方。例如，f(x) = 1/(x – 2) 的定义域是 x ≠ 2。对于 f(x) = √(x – 3)，我们需要 x ≥ 3。

To find the range, think about the output values that are possible. For f(x) = x² with domain -2

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