📚 Function Notation | 函数记号
A function is a rule that maps each input to exactly one output. In IGCSE Mathematics, mastering function notation is essential for solving equations, working with graphs, and tackling more advanced algebra. This article will guide you through f(x), evaluating functions, composite functions, and inverse functions – the key skills examined under the Edexcel specification.
函数是一种将每个输入映射到恰好一个输出的规则。在IGCSE数学中，掌握函数记号对于解方程、处理图像以及应对更高级的代数内容至关重要。本文将带你逐步理解 f(x)、函数求值、复合函数和反函数——这些正是Edexcel考纲中考查的核心技能。
1. What is a Function? | 什么是函数？
A function is a special relationship between two sets: the domain (set of inputs) and the range (set of outputs). For a rule to be a function, each input value must correspond to exactly one output value. Think of a function machine: you put a number in, a rule is applied, and a number comes out.
函数是两个集合之间的一种特殊关系：定义域（输入集合）和值域（输出集合）。对于一个规则，要成为函数，每个输入值必须恰好对应一个输出值。想象一个函数机器：你放入一个数，应用一条规则，然后输出一个数。
kes the shape of an equation like y = 3x + 2, but with a very important notational upgrade – using f(x) instead of y.
在代数形式中，函数通常呈现为类似 y = 3x + 2 的方程，但有一个非常重要的符号升级——用 f(x) 代替 y。
- A mapping where an input gives two different outputs is not a function.
- 一个输入给出两个不同输出的映射不是函数。
2. Function Notation: f(x) | 函数记号：f(x)
The notation f(x) is read as 'f of x' is the name of the function, x is the variable, and the expression 2x + 5 gives the rule.
记号 f(x) 读作“f of x”，表示当输入为 x 时函数 f 的输出。它并不意味着 f 乘以 x。例如，若我们定义 f(x) = 2x + 5，那么 f 是函数的名称，x 是变量，而表达式 2x + 5 给出了规则。
Other letters can also be used, such as g(x), h(x), or even v(t). In IGCSE, you will often see f and g combined.
其他字母也可以使用，如 g(x)、h(x) 甚至 v(t)。在IGCSE中，你将经常看到 f 和 g 组合出现。
3. Evaluating a Function | 函数求值
riable with the given number or expression and simplify. For f(x) = x² − 3, evaluate f(4):
要求函数值，只需将变量替换为给定的数或表达式并化简。对于 f(x) = x² − 3，计算 f(4)：
f(4) = (4)² − 3 = 16 − 3 = 13
o avoid sign errors. If a function is f(x) = 3x − 1, then f(−2) = 3(−2) − 1 = −6 − 1 = −7.
务必使用括号以避免符号错误。如果函数是 f(x) = 3x − 1，那么 f(−2) = 3(−2) − 1 = −6 − 1 = −7。
valuate a function for an algebraic expression. For the same f(x) = 3x − 1, find f(a + 2):
你也可以对代数式进行求值。对于同一个 f(x) = 3x − 1，求 f(a + 2)：
f(a + 2) = 3(a + 2) − 1 = 3a + 6 − 1 = 3a + 5
4. Solving f(x) = k | 解方程 f(x) = k
n equal to the output value and solve for x. For f(x) = 4x − 7, find x such that f(x) = 9.
有时你会被给出输出值，需要求输入值。将函数表达式设为等于输出值，然后解出 x。对于 f(x) = 4x − 7，求使得 f(x) = 9 的 x。
4x − 7 = 9 → 4x = 16 → x = 4
= 4, so x = 2 or x = −2. Remember that solving f(x) = k can yield more than one solution.
一旦将函数写出，这仅仅是解线性方程。对于像 f(x) = x² + 1 的二次函数，解 f(x) = 5 得到 x² = 4，所以 x = 2 或 x = −2。请记住，解 f(x) = k 可能产生多个解。
5. Composite Functions: fg(x) | 复合函数：fg(x)
another. The notation fg(x) means apply g first, then apply f to the result. It is read as 'f of g of x'. Do not confuse fg(x) with f(x) × g(x).
复合函数是一个接一个地应用函数。记号 fg(x) 表示先应用 g，然后将结果代入 f。它读作“f of g of x”。不要将 fg(x) 与 f(x) × g(x) 混淆。
If f(x) = 2x and g(x) = x + 3, then fg(x) = f(g(x)) = f(x + 3) = 2(x + 3) = 2x + 6.
如果 f(x) = 2x，g(x) = x + 3，那么 fg(x) = f(g(x)) = f(x + 3) = 2(x + 3) = 2x + 6。
gf(x) = g(f(x)) = g(2x) = 2x + 3. As you can see, fg(x) and gf(x) are generally different.
顺序很重要：gf(x) = g(f(x)) = g(2x) = 2x + 3。如你所见，fg(x) 和 gf(x) 通常是不同的。
| Order | 1st step | 2nd step | Result |
|---|---|---|---|
| fg(x) | Evaluate g(x) | Put into f | f(output of g) |
| gf(x) | Evaluate f(x) | Put into g | g(output of f) |
6. Finding Composite Functions Algebraically | 代数求复合函数
n of x in f(x). Always work from the inside out. Example: f(x) = x² − 1 and g(x) = 2x + 1.
要求出 fg(x) 的一般表达式，将整个 g(x) 代入到 f(x) 中 x 的位置。始终从内向外处理。示例：f(x) = x² − 1，g(x) = 2x + 1。
fg(x) = f(2x + 1) = (2x + 1)² − 1 = 4x² + 4x + 1 − 1 = 4x² + 4x
ite expression: fg(3) = 4(3)² + 4(3) = 36 + 12 = 48. Both approaches give the same result.
如果需要求数值，比如 fg(3)，你可以先计算 g(3)，再代入 f，或者将 3 代入复合表达式：fg(3) = 4(3)² + 4(3) = 36 + 12 = 48。两种方法结果一致。
7. Inverse Functions: f⁻¹(x) | 反函数：f⁻¹(x)
The inverse function reverses the effect of the original function. If f maps x to y, then f⁻¹ maps y back to x. The notation f⁻¹(x) is read as 'f inverse of x' and is not the same as a reciprocal.
反函数的作用是逆转原函数的效果。如果 f 将 x 映射到 y，那么 f⁻¹ 将 y 映射回 x。记号 f⁻¹(x) 读作“f inverse of x”，它不同于倒数。
A function must be one-to-one for its inverse to itself be a function, but at IGCSE level you will mainly work with linear functions and some simple quadratics with restricted domains.
一个函数必须是一一映射，其反函数本身才能是函数，但在IGCSE阶段，你主要处理线性函数以及一些限制定义域的简单二次函数。
8. Finding the Inverse Function | 求反函数
= f(x), swap x and y, then rearrange to make y the subject. The new expression is f⁻¹(x).
要从 f(x) 求 f⁻¹(x)，遵循以下步骤：写出 y = f(x)，交换 x 和 y，然后整理使 y 成为主项。新表达式就是 f⁻¹(x)。
Example: f(x) = 3x − 4.
示例：f(x) = 3x − 4。
Step 1: y = 3x − 4
Step 2: Swap x and y → x = 3y − 4
Step 3: Solve: x + 4 = 3y → y = (x + 4)/3
f⁻¹(x) = (x + 4)/3
t f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. For the above: f((x+4)/3) = 3((x+4)/3) − 4 = x+4−4 = x.
要检验你的反函数，验证 f(f⁻¹(x)) = x 以及 f⁻¹(f(x)) = x。就上例而言：f((x+4)/3) = 3((x+4)/3) − 4 = x+4−4 = x。
9. Function Machines and Number Pairs | 函数机器与数值对
nction machine is a visual way to understand inputs and outputs. If the machine is labeled '× 2 + 3' ds to f(x) = 2x + 3. The inverse machine performs the reverse operations in reverse order: '− 3 ÷ 2'. This matches f⁻¹(x) = (x − 3)/2.
函数机器是理解输入和输出的一种直观方式。如果机器标着“× 2 + 3”，那么输入 5 将输出 13。这对应于 f(x) = 2x + 3。它的反向机器则以相反的顺序执行逆运算：“− 3 ÷ 2”，这恰好是 f⁻¹(x) = (x − 3)/2。
At IGCSE, you may be asked to complete a table of values for a function and its inverse, using the logic of mapping backwards.
在IGCSE中，你可能会被要求用反向映射的逻辑填写一个函数及其反函数的数值表。
10. Common Mistakes and Exam Tips | 常见错误和考试技巧
Mistake 1: Thinking f(x) means f times x. Always treat it as an instruction. Mistake 2: Misreading fg(x) as gf(x). Always check the order—apply the second function first. Mistake 3: Forgetting to use brackets when evaluating f(−3) or f(a+b), leading to sign errors.
错误1：认为 f(x) 是 f 乘以 x。始终将它视为指令。错误2：将 fg(x) 误读为 gf(x)。务必检查顺序——先应用后写的函数。错误3：在求 f(−3) 或 f(a+b) 时忘记用括号，导致符号错误。
Mistake 4: When finding an inverse, not swapping x and y, or not making y the subject correctly. Practice writing each step neatly.
错误4：求反函数时，没有交换 x 和 y，或者未能正确将 y 表达为主项。练习工整地书写每一步。
Tip: In composite function questions, if you are asked to find fg(2), you can find g(2) numerically, then find f of that number. This often reduces algebra mistakes.
技巧：在复合函数题目中，如果要求 fg(2)，你可以先求出 g(2) 的数值，再求 f 作用于该数。这通常能减少代数错误。
11. Working with Quadratic and Other Functions | 处理二次函数及其他函数
unctions in later topics. Evaluating f(−2) for f(x) = 2x² + 3x gives 2(4) − 6 = 8 − 6 = 2.
Edexcel IGCSE 包含的内容不限于线性函数。你可能遇到 f(x) = x² + 1、g(x) = 1/x，甚至后续章节中的三角函数。对于 f(x) = 2x² + 3x，求 f(−2) 得到 2(4) − 6 = 8 − 6 = 2。
that the inverse is a function. For example, f(x) = x² for x ≥ 0 has the inverse f⁻¹(x) = √x. IGCSE questions usually specify the domain when required.
在求二次函数的反函数时，你需要考虑限制定义域，使得反函数也是一个函数。例如，f(x) = x²（x ≥ 0）的反函数为 f⁻¹(x) = √x。IGCSE 题目通常会在需要时明确指定定义域。
12. Summary and Practice | 总结与练习
fg(x), and derive f⁻¹(x). These skills form the backbone of many IGCSE algebra and graph questions.
函数记号是描述代数规则的一种简洁方式。确保自己能熟练地求 f(x) 的值、解 f(x) = k、组合函数得到 fg(x) 以及推导 f⁻¹(x)。这些技能构成了许多 IGCSE 代数和图像问题的根基。
Practice by writing your own function machines, checking inverse functions using composition, and working through past paper questions where function notation is tested alongside graph transformations or equations.
通过编写你自己的函数机器、利用复合函数检验反函数以及练习真题来进行训练，在这些真题中，函数记号常常与图像变换或方程一起考查。
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