📚 Functions (Edexcel A-Level Maths) | 函数（爱德思 A-Level 数学）
Functions are central to A-Level mathematics, linking algebraic expressions to graphical representations and modelling real-world situations. In this revision guide, we cover the key concepts required for Edexcel A-Level Maths, including domain and range, composite and inverse functions, the modulus function, transformations and more.
函数是 A-Level 数学的核心内容，它将代数表达式与图像表示联系起来，并用于模拟现实世界的情形。在本复习指南中，我们将梳理 Edexcel A-Level 数学的重要概念，包括定义域与值域、复合函数与反函数、绝对值函数、图像变换等内容。
1. What is a Function? | 什么是函数？
nput value exactly one output value. If we name the function f, we write y = f(x), read as "f of x" mple, f(x) = x² + 3 is a function: every real number x produces exactly one value of f(x).
函数通常记作 f(x)，是一个规则，为每个输入值 x 分配恰好一个输出值 y。我们写作 y = f(x)。例如，f(x) = x² + 3 是一个函数。所有允许的输入值构成定义域，实际产生的输出值构成值域。
function. A circle such as x² + y² = 25 fails this test, because a vertical line can cut it twice.
判断一个图像是否表示函数的一个简单方法是垂直线检验：如果任何垂直线与图像相交超过一次，则它不是函数。例如圆 x² + y² = 25 就无法通过检验，因为一条垂直线可以与它相交两次。
Functions are usually described as one-to-one (each output comes from exactly one input) or many-to-one (several inputs may share the same output). Both are valid functions, but only one-to-one functions have an inverse function that is also a function.
函数通常分为一对一（每个输出只来自一个输入）与多对一（多个输入可能对应同一个输出）。两者都是合法的函数，但只有一对一函数才存在仍是函数的反函数。
2. Domain and Range | 定义域与值域
ominators not being allowed to equal zero and expressions under a square root being non-negative. For example, f(x) = √x - 2 has domain x ≥ 2, or {x ∈ ℝ : x ≥ 2}.
函数的定义域是使函数有定义的所有可能输入值（x 值）的完整集合。常见的限制包括分母不为零以及平方根下的表达式非负。例如，f(x) = √x - 2 的定义域为 x ≥ 2，或记作 {x ∈ ℝ : x ≥ 2}。
. For f(x) = x² - 4x + 7, completing the square gives f(x) = (x - 2)² + 3, so the minimum value is 3 and the range is f(x) ≥ 3.
值域是函数所能产生的所有输出值（y 值）的集合。为了求出值域，通常需要考虑图像的形状或通过配方法。对于 f(x) = x² - 4x + 7，配方得 f(x) = (x - 2)² + 3，因此最小值为 3，值域为 f(x) ≥ 3。
Three standard examples you should know instantly are listed below.
以下三个标准例子应当能够立即写出。
- f(x) = x² + 5: domain ℝ, range f(x) ≥ 5
- f(x) = 1/(x - 3): domain x ≠ 3, range f(x) ≠ 0
- f(x) = √x: domain x ≥ 0, range f(x) ≥ 0
f(x) = x² + 5：定义域 ℝ，值域 f(x) ≥ 5；f(x) = 1/(x - 3)：定义域 x ≠ 3，值域 f(x) ≠ 0；f(x) = √x：定义域 x ≥ 0，值域 f(x) ≥ 0。
| Function | Domain | Range |
|---|---|---|
| f(x) = 1/x | x ≠ 0 | f(x) ≠ 0 |
| f(x) = √x | x ≥ 0 | f(x) ≥ 0 |
| f(x) = eˣ | x ∈ ℝ | f(x) > 0 |
| f(x) = ln x | x > 0 | f(x) ∈ ℝ |
The table above summarises typical domains and ranges for common functions. Notice that for eˣ the output is always positive, while for ln x the input must be strictly greater than zero.
上表总结了常见函数的典型定义域与值域。注意 eˣ 的输出恒为正，而 ln x 的输入必须严格大于零。
3. Set and Interval Notation | 集合记法与区间记法
xcel A-Level, you must be able to define domains and ranges using set notation, such as {x ∈ ℝ : x ≥ 0}. The symbol ∈ means "is an element of", ℝ denotes the set of real numbers, and the colon is read as "such that".
在爱德思 A-Level 中，你需要能够使用集合表示法来定义定义域与值域，例如 {x ∈ ℝ : x ≥ 0}。符号 ∈ 表示 "属于"，ℝ 表示实数集，冒号读作 "使得"。
eans it is excluded: [0, ∞) is the same as x ≥ 0, while (2, 5] means 2 < x ≤ 5. The symbol ∪ denotes union (either set) and ∩ denotes intersection (both sets).
区间记法是更简洁的替代写法。方括号表示端点包含在内，圆括号表示端点不包含：[0, ∞) 等价于 x ≥ 0，而 (2, 5] 表示 2 < x ≤ 5。符号 ∪ 表示并集（属于任一集合），∩ 表示交集（同时属于两个集合）。
{x ∈ ℝ : x > 3} = (3, ∞) {x ∈ ℝ : -1 ≤ x < 4} = [-1, 4)
When you combine two domains, for example when adding functions, the result is the intersection of the two domains, not the union.
当你把两个函数组合时（例如相加），所得函数的定义域是两个定义域的交集，而不是并集。
4. Composite Functions | 复合函数
f(g(x)). Note the order: g acts first, then f, so we always read from the inside outwards.
复合函数是把一个函数作用于另一个函数的结果。复合函数 fg 定义为 fg(x) = f(g(x))。注意顺序：先作用 g，再作用 f，即由内向外计算。
) = 2(x² + 1) - 3 = 2x² - 1. These two answers are different, so in general fg(x) ≠ gf(x).
设 f(x) = x² + 1，g(x) = 2x - 3，则 fg(x) = f(2x - 3) = (2x - 3)² + 1 = 4x² - 12x + 10。交换顺序得 gf(x) = g(x² + 1) = 2(x² + 1) - 3 = 2x² - 1。两者不同，因此一般来说 fg(x) ≠ gf(x)。
es of x in the domain of g for which g(x) lies in the domain of f. For instance, if g(x) = √x and f(x) = 1/(x - 2), then fg(x) = 1/(√x - 2) requires x ≥ 0 and √x ≠ 2, that is x ≥ 0 with x ≠ 4.
复合函数 fg 只对满足以下条件的 x 存在：x 属于 g 的定义域，且 g(x) 落在 f 的定义域内。例如 g(x) = √x，f(x) = 1/(x - 2)，则 fg(x) = 1/(√x - 2) 要求 x ≥ 0 且 √x ≠ 2，即 x ≥ 0 且 x ≠ 4。
5. Inverse Functions | 反函数
x in the domain of f⁻¹. An inverse function exists only if f is one-to-one on its domain.
反函数 f⁻¹ 的作用与 f 相反。它满足 f⁻¹(f(x)) = x（对 f 定义域内的每个 x），以及 f(f⁻¹(x)) = x（对 f⁻¹ 定义域内的每个 x）。只有当 f 在其定义域上是一对一时，反函数才存在。
to make x the subject, then swap x and y. For f(x) = x² + 3 with x ≥ 0: y = x² + 3, so x = √y - 3, and therefore f⁻¹(x) = √x - 3 x ≥ 0 was essential, because without it f would be many-to-one and would have no inverse.
求反函数的方法是：先写 y = f(x)，整理成以 x 为主元，再交换 x 与 y。对于 f(x) = x² + 3，x ≥ 0：由 y = x² + 3 得 x = √y - 3，故 f⁻¹(x) = √x - 3，定义域为 x ≥ 3。这里的限制 x ≥ 0 必不可少，否则 f 是多对一的，就没有反函数。
ntly the domain of f⁻¹ equals the range of f, and the range of f⁻¹ equals the domain of f.
y = f⁻¹(x) 的图像是 y = f(x) 的图像关于直线 y = x 的对称图形。因此 f⁻¹ 的定义域等于 f 的值域，f⁻¹ 的值域等于 f 的定义域。
domain(f⁻¹) = range(f) range(f⁻¹) = domain(f)
6. The Modulus Function | 绝对值函数
The modulus function |x| gives the distance of x from zero. Formally, |x| = x when x ≥ 0, and |x| when x < 0. Its graph is V-shaped with its vertex at the origin, and it is never negative.
绝对值函数 |x| 表示 x 到零点的距离。正式地，当 x ≥ 0 时 |x| = x；当 x < 0 时 |x| = -x。它的图像是以原点为顶点的 V 形，函数值永不为负。
To solve |2x - 3| s x = 4, and 2x - 3 = -5 gives x = -1. Always check both answers in the original equation.
解 |2x - 3| = 5 时，需分两种情况：2x - 3 = 5 得 x = 4，2x - 3 = -5 得 x = -1。务必把两个答案代回原方程检验。
For inequalities, |x - 1| < 3 becomes -3 < x - 1 < 3, so -2 < x < 4. A useful exam tip is that |a| < b means -b < a < b, whereas |a| > b means a < -b or a > b.
对于不等式，|x - 1| < 3 可化为 -3 < x - 1 < 3，即 -2 < x < 4。一个实用的考试技巧是：|a| < b 等价于 -b < a < b，而 |a| > b 等价于 a < -b 或 a > b。
The graph of y = |f(x)| btained from y = f(x) by reflecting any part below the x-axis upwards. The graph of y = f(|x|) keeps the right-hand side and reflects it into the left-hand side.
y = |f(x)| 的图像可由 y = f(x) 得到：把 x 轴下方的部分翻折到上方。y = f(|x|) 的图像则保留右半部分，并把它对称到左侧。
7. Transformations of Graphs | 图像变换
Transformations allow you to sketch unfamiliar graphs quickly from known ones. Changes made inside the bracket affect the x-direction and work in the opposite sense to what you might expect; changes made outside affect the y-direction and work as expected.
图像变换让你能由已知图像快速画出陌生函数的图像。括号内的改动影响 x 方向，且效果与直觉相反；括号外的改动影响 y 方向，效果与直觉一致。
| Transformation | Effect on graph |
|---|---|
| y = f(x) + a | translation by a units upwards |
| y = f(x + a) | translation by a units to the left |
| y = af(x) | vertical stretch, scale factor a |
| y = f(ax) | horizontal stretch, scale factor 1/a |
| y = -f(x) | reflection in the x-axis |
| y = f(-x) | reflection in the y-axis |
English description above, Chinese equivalent below for quick recall.
上表为英文对照，下面用中文再归纳一遍，方便快速记忆。
- y = f(x) + a：向上平移 a 个单位
- y = f(x + a)：向左平移 a 个单位
- y = af(x)：纵向拉伸 a 倍
- y = f(ax)：横向压缩为原来的 1/a
- y = -f(x)：关于 x 轴翻折
- y = f(-x)：关于 y 轴翻折
8. Mappings and Types of Mapping | 映射与映射类型
A mapping describes how elements of one set are matched with elements of another. Four cases are possible, and only two of them are functions.
映射描述一个集合的元素如何与另一个集合的元素对应。共有四种情况，其中只有两种属于函数。
| Type | A function? | Has inverse? |
|---|---|---|
| One-to-one | Yes | Yes |
| Many-to-one | Yes | No |
| One-to-many | No | No |
| Many-to-many | <
