📚 2.12 Functions – Notation, Composite, Inverse & Piecewise | 2.12 函数 – 记号、复合、逆与分段函数
Functions are one of the most powerful ideas in IGCSE Mathematics. They describe relationships between inputs and outputs, and the Edexcel 2.12 topic brings together notation, domain, range, composite functions, inverse functions and piecewise definitions. Mastering these concepts will not only boost your exam performance but also build a strong foundation for A Level Maths.
函数是 IGCSE 数学中最强大的思想之一。它们描述输入与输出之间的关系，Edexcel 2.12 专题将记号、定义域、值域、复合函数、逆函数和分段定义集中在一起。掌握这些概念不仅能提升你的考试成绩，还能为 A Level 数学打下扎实的基础。
1. What is a Function? | 什么是函数？
A function is a rule that assigns exactly one output to every input. Think of it as a number machine: you put a number in, the machine does something to it, and you get a single result out. For a relationship to be a function, each input cannot produce more than one output.
函数是一种规则，每个输入恰好对应一个输出。可以把它想象成一台数字机器：你放入一个数，机器对它进行某种操作，然后你得到一个唯一的结果。要使对应关系成为函数，每个输入不能产生多于一个输出。
We commonly denote inputs by x and outputs by y or f(x). The set of all possible inputs is the domain, and the set of all outputs that actually occur is the range.
我们通常用 x 表示输入，用 y 或 f(x) 表示输出。所有可能输入的集合是定义域，而实际出现的所有输出的集合是值域。
2. Function Notation – f(x) | 函数记号 – f(x)
tion, and the expression in brackets tells us the variable. This notation makes it easy to evaluate the function at different numbers, and it prepares you for composite and inverse work.
我们不再写 y = 2x + 3，而是写成 f(x) = 2x + 3。字母 f 是函数的名称，括号里的表达式告诉我们是哪个变量。这种记号让你能轻松地求出函数在不同数值处的取值，也为复合函数和逆函数的学习做好了准备。
If a question uses g(x) = x² – 5, then g(4) simply means ‘replace x with 4’. You get 4² – 5 = 11. Always be comfortable with different letters; the notation works the same way.
如果题目给出 g(x) = x² – 5，那么 g(4) 就表示“用 4 代替 x”。你得到 4² – 5 = 11。要习惯不同的字母，记号的使用方式完全相同。
3. Evaluating Functions | 求函数值
at a given value, substitute the number for every instance of the variable. For f(x) = 3x – 5, f(–2) = 3(–2) – 5 = –6 – 5 = – uestions you may also need to evaluate functions that contain fractions, squares or roots.
为了求函数在某个给定值处的取值，将数字代入每一个变量。对于 f(x) = 3x – 5，f(–2) = 3(–2) – 5 = –6 – 5 = –11。在 IGCSE 考题中，你还需要求包含分数、平方或根式的函数的值。
ion is given inside the brackets, treat it in exactly the same way. For h(x) = 2x + 1, h(a – 3) = 2(a – 3) + 1 = 2a – 6 + 1 = 2a – 5. The algebra remains straightforward if you work step by step.
当括号内是一个表达式时，也用同样的方法处理。对于 h(x) = 2x + 1，h(a – 3) = 2(a – 3) + 1 = 2a – 6 + 1 = 2a – 5。只要一步一步进行，代数运算依然很简单。
4. Domain of a Function | 函数的定义域
The domain of a function is the complete set of input values for which the function is defined. IGCSE problems often ask you to state the largest possible domain. You must watch out for two main restrictions: division by zero and square roots of negative numbers.
函数的定义域是函数有定义的全体输入值的集合。IGCSE 题目经常要求你写出最大可能的定义域。你必须留意两类主要限制：除以零和负数的平方根。
For f(x) = 1/(x – 2), x cannot be 2, so the domain is all real numbers except 2, often written as x ∈ ℝ, x ≠ 2. For g(x) = √x + 5, the expression inside the root must be ≥ 0, so x + 5 ≥ 0 ⇒ x ≥ –5.
对于 f(x) = 1/(x – 2)，x 不能等于 2，因此定义域是所有不等于 2 的实数，通常写作 x ∈ ℝ, x ≠ 2。对于 g(x) = √x + 5，根号内的表达式必须 ≥ 0，所以 x + 5 ≥ 0 ⇒ x ≥ –5。
5. Range of a Function | 函数的值域
s y ≥ 1. For linear functions without restrictions, the range is usually all real numbers.
值域是函数实际能产生的所有输出值的集合。对于二次函数如 f(x) = x² + 1，最小值是 1（当 x = 0 时），因此值域为 y ≥ 1。对于没有限制的线性函数，值域通常是全体实数。
main is deliberately restricted, which changes the range. If f(x) = x² is defined only for – ange comes from the smallest and largest outputs: f(0) = 0 and f(2) = 4, giving 0 ≤ y ≤ 4.
有时题目故意限制定义域，这会改变值域。如果 f(x) = x² 仅在 –1 ≤ x ≤ 2 上定义，值域由最小和最大输出决定：f(0) = 0，f(2) = 4，得出 0 ≤ y ≤ 4。
Always test critical points like the ends of domain intervals and any stationary points when finding the range.
求值域时，总是要检查区间端点以及任何驻点（极值点）处的取值。
6. Composite Functions | 复合函数
CSE, you must be able to form and simplify fg(x) and gf(x), and understand they are usually different.
当一个函数接在另一个函数之后使用时，就构成了复合函数。记号 fg(x) 表示先对 x 使用 g，然后对结果使用 f。注意顺序：fg(x) = f(g(x))。在 IGCSE 中，你必须能够建立并化简 fg(x) 和 gf(x)，并理解它们通常不相同。
en fg(x) = f(x²) = 3x² + 2, while gf(x) = g(3x + 2) = (3x + 2)². The order matters hugely.
例如：设 f(x) = 3x + 2，g(x) = x²。那么 fg(x) = f(x²) = 3x² + 2，而 gf(x) = g(3x + 2) = (3x + 2)²。顺序至关重要。
You may also need to evaluate a composite function at a specific number, like gf(4). Work from the inside out: first calculate f(4), then apply g to that number.
你可能还需要求复合函数在某个特定数值处的值，例如 gf(4)。从内向外计算：先求出 f(4)，然后把那个数代入 g。
7. Inverse Functions | 逆函数
cally, write y = f(x), swap x and y, and then solve for y. Finally, replace y with f⁻¹(x).
逆函数会逆转原函数的效果。如果 f(x) 把数 a 映射到 b，那么逆函数 f⁻¹(x) 就把 b 映射回 a。代数上求逆函数的方法是：写出 y = f(x)，交换 x 和 y，然后解出 y，最后把 y 换成 f⁻¹(x)。
For example, f(x) = 4x – 7 → y = 4x – 7 → swap: x = 4y – → solve: 4y = x + 7 → y = (x + 7)/4 → f⁻¹(x) = (x + 7)/4. Always check that f(f⁻¹(x)) = x.
例如，f(x) = 4x – 7 → y = 4x – 7 → 交换：x = 4y – 7 → 求解：4y = x + 7 → y = (x + 7)/4 → f⁻¹(x) = (x + 7)/4。务必检验 f(f⁻¹(x)) = x。
ut. The graphs of y = f(x) and y = f⁻¹(x) are reflections of each other in the line y = x.
仅当原函数是一对一（单射）时，逆函数才存在，即每个输出只来自一个输入。y = f(x) 和 y = f⁻¹(x) 的图像关于直线 y = x 对称。
8. Piecewise Functions | 分段函数
sub-function and its condition. For instance, f(x) = { x² for x < 0; 2x + 1 for 0 ≤ x ≤ 3; – > 3 }. You must be able to sketch these graphs and evaluate the function at given numbers.
分段函数对 x 的不同区间使用不同的规则。它用一个大括号列出每一段子函数及其条件。例如，f(x) = { x² 当 x < 0； 2x + 1 当 0 ≤ x ≤ 3； –x + 5 当 x > 3 }。你必须会画出这些函数的草图，并会求出给定数值处的函数值。
unction above, f(2) uses the middle branch: 2(2) + 1 = 5, while f(4) uses the last branch: –4 + 5 = 1.
求值时，只需判断输入落在哪个区间，并使用对应的公式。对于上面的分段函数，f(2) 用到中间那一支：2(2) + 1 = 5，而 f(4) 用到最后一支：–4 + 5 = 1。
9. Graphs of Functions | 函数的图像
nderstand domain and range visually. For example, the minimum point of a quadratic reveals the start of the range.
IGCSE 考生应当能熟练画出线性、二次、反比例和分段函数的草图。了解 y = f(x) 图像的形状有助于你直观地理解定义域和值域。例如，二次函数的最低点揭示了值域的起点。
graph across it. If you are asked to read values from a graph of a composite function, remember the inside-out order.
画逆函数图像时，记住对称性质：画出直线 y = x，然后以它为对称轴映射原函数的图像。如果要求从复合函数的图像中读取数值，记住由内向外的计算顺序。
10. Common Mistakes & How to Avoid Them | 常见错误及避免方法
One of the most frequent errors is mixing up fg(x) and gf(x). Always read the notation carefully: fg means ‘g first’. Write out the inner substitution explicitly before applying the outer function.
最常见的错误之一是混淆 fg(x) 和 gf(x)。一定要仔细阅读记号：fg 表示“先做 g”。在应用外层函数之前，要先把内层的代换写清楚。
Another pitfall is forgetting to state the domain when giving an inverse function. If the original function has a restricted domain, the inverse's domain is the original's range, and you should write it down.
另一个陷阱是给出逆函数时忘记声明定义域。如果原函数的定义域有限制，那么逆函数的定义域就是原函数的值域，你应该把它写出来。
xpressions like f(x + h), students often forget to apply every operation. Treat f(x) = 2x² – x, then f(x + h) = 2(x + h)² – (x + h), and expand fully. Rushing leads to careless mistakes.
当化简像 f(x + h) 这样的表达式时，学生常常忘记应用所有运算。例如 f(x) = 2x² – x，那么 f(x + h) = 2(x + h)² – (x + h)，并且要完全展开。急躁会导致粗心错误。
11. Summary – Linking All the Pieces | 总结 – 串联所有知识
The 2.12 functions topic ties together several skills: reading and using f(x) notation, manipulating algebraic expressions, solving equations, and understanding graphs. In the exam, you might be asked to form a composite, find its value, then determine its inverse, all within a single question. Confidence comes from practicing each layer separately and then combining them.
2.12 函数专题将多项技能融合在一起：阅读和使用 f(x) 记号、处理代数式、解方程以及理解图像。在考试中，可能会要求你构造复合函数、求其值、再确定其逆函数，全在同一道题里。信心来自先分别练习每一层技能，然后再把它们结合起来。
Use a systematic approach: identify the function, check the domain, work from the inside out for composites, and swap-and-solve for inverses. With enough revision, functions will become one of the most predictable and rewarding parts of your IGCSE paper.
采用系统的方法：识别函数，检查定义域，复合函数从内向外计算，逆函数通过交换并求解。经过充分的复习，函数将成为你 IGCSE 试卷中最可预测且最有收获的部分之一。
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