📚 Understanding Functions: Concept and Notation | 函数的概念与表示
Functions are one of the most fundamental ideas in mathematics. In the Edexcel IGCSE syllabus, mastering the concept and notation of functions unlocks every later topic, from graphs and equations to trigonometry and calculus. This guide explains exactly what a function is, how to write it, and how to work with it confidently in your exams.
函数是数学中最基本的概念之一。在 Edexcel IGCSE 考纲中，掌握函数的概念与表示是学习图像、方程、三角学和微积分等后续内容的钥匙。本指南将准确讲解什么是函数、如何表示函数，以及如何在考试中自信地运用函数。
1. What Is a Function? | 什么是函数？
A function is a rule that assigns exactly one output to each valid input. Think of it like a machine: you put an input value in, the machine applies the rule, and exactly one output comes out.
函数是一种规则，它将每个合法的输入值唯一地对应到一个输出值。你可以把它想象成一台机器：投入一个输入值，机器执行规则，恰好产生一个输出值。
Three key conditions define a function:
函数有三个关键条件：
- Every input in the domain must produce an output.
- 每个定义域内的输入都必须产生一个输出。
- The same input always produces the same output (deterministic).
- 相同的输入总是产生相同的输出（确定性）。
- No input can produce two different outputs.
- 任何输入都不能产生两个不同的输出。
Input x → Function f → Output f(x)
A simple example: the rule “double the input and add 3” is a function, because any number you choose gives exactly one result.
一个简单例子：”将输入加倍再加 3″ 这个规则就是一个函数，因为任何你选择的数都恰好产生一个结果。
2. Function Notation f(x) | 函数记号 f(x)
In IGCSE mathematics, we write functions using the notation f(x), read as “f of x”. The letter f is the name of the function, x is the input variable, and the whole symbol f(x) represents the output.
在 IGCSE 数学中，我们用记号 f(x) 表示函数，读作”f of x”。字母 f 是函数的名称，x 是输入变量，整个符号 f(x) 代表输出值。
For example, the function f(x) = 2x + 3 tells you: take the input x, multiply it by 2, then add 3. When we write f(5), we mean the output when x = 5:
例如，函数 f(x) = 2x + 3 表示：将输入 x 乘以 2，再加上 3。当我们写 f(5) 时，意思是 x = 5 时的输出：
f(5) = 2(5) + 3 = 10 + 3 = 13
You can name functions with different letters: g(x), h(x), or F(x) are all common. Choose a letter that helps you remember what the function does.
你可以用不同的字母来命名函数：g(x)、h(x) 或 F(x) 都是常见的。选择一个能帮助你记住函数功能的字母。
3. Domain, Codomain and Range | 定义域、陪域与值域
Three sets are essential when describing a function fully: the domain, the codomain and the range.
完整描述一个函数需要三个重要的集合：定义域、陪域和值域。
- The domain is the set of all possible input values for which the function is defined.
- 定义域是函数有定义的所有可能输入值的集合。
- The range is the set of all output values that the function actually produces.
- 值域是函数实际产生的所有输出值的集合。
- The codomain is the larger set of all outputs that the rule could possibly produce.
- 陪域是规则可能产生的所有输出的更大集合。
For f(x) = x² with domain ℝ (all real numbers), the range is [0, ∞), since squaring any real number never gives a negative result.
对于 f(x) = x²，若定义域为 ℝ（全体实数），则值域为 [0, ∞)，因为任何实数的平方都不会是负数。
| Function | Domain | Range |
| f(x) = x² | ℝ | y ≥ 0 |
| f(x) = 1/x | x ≠ 0 | y ≠ 0 |
| f(x) = √x | x ≥ 0 | y ≥ 0 |
Always identify the domain first when analysing a function, because the range depends entirely on which inputs are allowed.
在分析函数时要先确定定义域，因为值域完全取决于允许哪些输入值。
4. Evaluating Functions | 函数的求值
To evaluate a function at a specific input, substitute that value directly into the expression. This is a core skill tested frequently in IGCSE papers.
要在特定输入处求函数值，只需将该值直接代入表达式中。这是 IGCSE 考试中反复考查的核心技能。
Example 1: f(x) = 3x – 1, find f(2).
例 1：f(x) = 3x – 1，求 f(2)。
f(2) = 3(2) – 1 = 6 – 1 = 5
Example 2: f(x) = x² + 2x, find f(-3). Use brackets for negative inputs!
例 2：f(x) = x² + 2x，求 f(-3)。代入负数时一定要加括号！
f(-3) = (-3)² + 2(-3) = 9 – 6 = 3
Common mistake: writing -3² = -9 instead of (-3)² = 9. The brackets tell you to square the negative number itself, not just the digit 3.
常见错误：写成 -3² = -9 而非 (-3)² = 9。括号表示要对负数本身进行平方，而不仅仅是数字 3。
5. One-to-One and Many-to-One Functions | 一对一与多对一函数
Functions fall into two important categories: one-to-one and many-to-one. This distinction decides whether a function has an inverse.
函数分为两个重要类别：一对一函数和多对一函数。这种区分决定了函数是否存在反函数。
- A one-to-one function maps each distinct input to a distinct output.
- 一对一函数将每个不同的输入映射到不同的输出。
- A many-to-one function maps different inputs to the same output.
- 多对一函数将不同的输入映射到相同的输出。
Example: f(x) = x² is many-to-one, because f(2) = 4 and f(-2) = 4. Both inputs 2 and -2 produce the same output 4.
例子：f(x) = x² 是多对一函数，因为 f(2) = 4 且 f(-2) = 4。输入 2 和 -2 都产生相同的输出 4。
Use the vertical line test on a graph: if any vertical line crosses the curve more than once, the curve is not a function. Use the horizontal line test to check one-to-one: if any horizontal line crosses more than once, the function is many-to-one.
在图像上使用竖线检验：若任何竖直线与曲线相交超过一次，则该曲线不是函数。用水平线检验判断一对一：若任何水平线相交超过一次，则该函数是多对一函数。
6. Composite Functions | 复合函数
A composite function combines two functions by applying one after the other. The notation fg(x) means f(g(x)): apply g first, then feed the result into f.
复合函数通过依次应用两个函数来组合它们。记号 fg(x) 表示 f(g(x))：先应用 g，再将结果代入 f。
Work systematically from the inside out. Example: f(x) = 2x and g(x) = x + 1.
按照从内向外的顺序系统计算。例子：f(x) = 2x，g(x) = x + 1。
Step 1: Apply g to x → g(x) = x + 1.
第一步：对 x 应用 g → g(x) = x + 1。
Step 2: Feed g(x) into f → f(g(x)) = 2(x + 1) = 2x + 2.
第二步：将 g(x) 代入 f → f(g(x)) = 2(x + 1) = 2x + 2。
fg(x) = 2x + 2, while gf(x) = g(2x) = 2x + 1
Important: In general fg(x) ≠ gf(x). The order of composition matters! Always check which function is applied first.
重要：一般情况下 fg(x) ≠ gf(x)。复合的顺序很重要！始终确认先应用哪个函数。
7. Inverse Functions | 反函数
The inverse function, written f⁻¹(x), reverses the effect of the original function. If f(a) = b, then f⁻¹(b) = a. Only one-to-one functions have inverses.
反函数，记作 f⁻¹(x)，逆转原函数的效果。若 f(a) = b，则 f⁻¹(b) = a。只有一对一函数才有反函数。
To find an inverse function, follow three steps:
求反函数只需三步：
- Write the function as y = f(x).
- 写出 y = f(x)。
- Swap x and y in the equation.
- 在方程中交换 x 和 y。
- Solve the new equation for y; this gives f⁻¹(x).
- 解新方程得到 y，即为 f⁻¹(x)。
Example: find the inverse of f(x) = 2x + 3.
例子：求 f(x) = 2x + 3 的反函数。
y = 2x + 3 → x = 2y + 3 → 2y = x – 3 → y = (x – 3)/2
Therefore f⁻¹(x) = (x – 3)/2. Check the property:
因此 f⁻¹(x) = (x – 3)/2。验证性质：
f⁻¹(f(x)) = f⁻¹(2x + 3) = ((2x + 3) – 3)/2 = x ✔
The domain of f⁻¹ equals the range of f, and vice versa. State this when a question asks about domains.
f⁻¹ 的定义域等于 f 的值域，反之亦然。当题目问及定义域时，请指明这一点。
8. Graphs of Functions | 函数图像
The graph of y = f(x) is the set of all points (x, f(x)) plotted on the coordinate plane. Reading graphs correctly is a major IGCSE skill.
y = f(x) 的图像是坐标平面上所有点 (x, f(x)) 的集合。正确读图是 IGCSE 的一项重要技能。
- To find the x-intercepts, set f(x) = 0 and solve for x.
- 求 x 截距：令 f(x) = 0，解出 x
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