📚 A-Level Mathematics: Basic Concepts and Representations of Functions | A-Level 数学：函数的基本概念与表示
A function is one of the most important ideas in A-Level mathematics. It describes a special relationship between two sets, where each input gives exactly one output. Understanding the formal definition, notation and different ways to represent a function is essential for solving problems in algebra, calculus and many other topics.
函数是 A-Level 数学中最重要的概念之一。它描述两个集合之间的一种特殊关系：每一个输入都有唯一确定的输出。理解函数的正式定义、记法以及不同的表示方式，是在代数、微积分及其他众多主题中解题的基础。
1. What Is a Function? | 什么是函数？
In mathematics, a function is a rule that assigns to every element x in a set called the domain exactly one element y in a set called the range. This condition of uniqueness is the key difference between a function and a general relation.
在数学中，函数是一种规则：对于定义域中的每一个元素 x，它都有且仅有一个对应的值 y 属于值域。这种唯一性条件正是函数与一般关系的本质区别。
We write f: x → y. If the input is 3 and the output is 9 under the rule “square the input”, then f(3) = 9. The arrow notation and the f(x) notation are both used throughout the A-Level course.
我们记作 f: x → y。如果输入为 3，且规则是“将输入平方”，那么输出为 9，即 f(3) = 9。箭头记法 f: x → y 与 f(x) 记法在整个 A-Level 课程中都会用到。
2. Domain, Codomain and Range | 定义域、陪域与值域
The domain is the set of all possible input values. The codomain is the set of all possible output values that the function could produce. The range is the subset of the codomain that is actually produced by the function.
定义域是所有可能输入值的集合。陪域是函数可能产生的所有输出值的集合。值域是陪域中函数实际产生的那些值组成的子集。
For example, for f(x) = x², if the domain is all real numbers ℝ, the codomain is ℝ, but the range is only non-negative real numbers, because a square is never negative. We write range = {y ∈ ℝ : y ≥ 0}.
例如，对于 f(x) = x²，若定义域为全体实数 ℝ，陪域为 ℝ，但值域仅为非负实数，因为平方永远不会为负数。我们写作 range = {y ∈ ℝ : y ≥ 0}。
In A-Level questions, always state the domain of a function first. If a domain is not given, you may assume the largest possible domain for which the function is defined, such as x ≠ 0 for f(x) = 1/x.
在 A-Level 题目中，永远先给出函数的定义域。若题目没有给出定义域，则通常取使函数有意义的最大定义域，例如对 f(x) = 1/x，要求 x ≠ 0。
3. Function Notation and Evaluation | 函数记法与求值
The notation f(x) is read as “f of x”. It represents the value of f at the input x. To evaluate f(2), replace each x in the formula by 2 and simplify.
记法 f(x) 读作 “f 在 x 处的值”，它表示函数 f 在输入 x 时的取值。要求 f(2)，只需把公式中的每个 x 都替换为 2，然后化简即可。
Example: if f(x) = 3x² – 2x + 1, then f(2) = 3(2)² – 2(2) + 1 = 12 – 4 + 1 = 9. Substitution is the most basic skill in working with functions.
例如：若 f(x) = 3x² – 2x + 1，则 f(2) = 3(2)² – 2(2) + 1 = 12 – 4 + 1 = 9。代入法是最基本的函数运算技能。
You may also see functions written like y = x² + 1. Here y is the output. The notation f(x) does not change the rule; it simply gives the function a name.
你也会看到函数写作 y = x² + 1。其中 y 是输出值。f(x) 这种记法并不改变函数规则，只是给函数起一个名字。
4. One-to-One and Many-to-One Functions | 一一映射与多一映射
A function is one-to-one (injective) if different inputs always produce different outputs. For example, f(x) = 2x + 3 is one-to-one, because x₁ ≠ x₂ means f(x₁) ≠ f(x₂).
如果一个函数中，不同的输入必然产生不同的输出，则称该函数是一一映射（单射）。例如 f(x) = 2x + 3 就是一一映射，因为 x₁ ≠ x₂ 可推出 f(x₁) ≠ f(x₂)。
A many-to-one function can produce the same output from different inputs. For example, f(x) = x² is many-to-one because f(-2) = 4 and f(2) = 4.
多一映射函数是指不同的输入可能产生相同的输出。例如 f(x) = x² 是多一映射，因为 f(-2) = 4 且 f(2) = 4。
For an inverse function to exist, the original function must be one-to-one. Many-to-one functions require a restricted domain before an inverse can be defined.
若要存在反函数，原函数必须是一一映射。对于多一映射函数，必须先限制定义域，才能定义反函数。
5. Graphical Representation and the Vertical Line Test | 图像表示与垂线检验
A function can be represented by a curve on a graph. The horizontal axis shows the input values (x), and the vertical axis shows the output values (y).
函数可以用坐标系中的曲线来表示。横轴表示输入值 x，纵轴表示输出值 y。
The vertical line test is a quick way to decide whether a graph represents a function: if any vertical line intersects the graph more than once, then the graph is not a function.
垂线检验是判断图像是否表示函数的快速方法：如果任意一条竖直线与图像相交超过一次，则该图像不是函数。
For example, a circle fails this test because a vertical line cuts through it twice, while the graph of y = x² passes it. In A-Level, you must also identify domain and range from the graph by looking at the extent of the curve on the axes.
例如，圆会失败该检验，因为竖直线会与其相交两次；而 y = x² 的图像则能通过。在 A-Level 中，你还需要通过观察曲线在坐标轴上的范围来从图像中判断定义域和值域。
6. Algebraic Representation | 代数表示
The most common way to represent a function is by an algebraic formula, such as f(x) = 2x – 1, f(x) = eˣ, or f(x) = sin x. These formulas are compact and allow exact calculations.
最常见的函数表示方式是代数公式，如 f(x) = 2x – 1、f(x) = eˣ 或 f(x) = sin x。这些公式简洁，而且可以进行精确计算。
Some functions are defined implicitly by equations such as x² + y² = 1. In A-Level, however, we usually work with explicit form y = f(x) to make evaluation and differentiation simpler.
有些函数由方程隐式定义，例如 x² + y² = 1。然而在 A-Level 中，我们通常使用显式形式 y = f(x)，这样求值和求导都更方便。
You should be able to switch between verbal rules, algebraic formulas, tables of values and graphs. This flexibility is called using multiple representations of a function.
你应该能在文字规则、代数公式、数值表格和图像之间相互转换。这种灵活性被称为函数的多种表示法。
7. Mapping Diagrams and Sets | 映射图与集合
A mapping diagram shows two sets with arrows connecting each input to its output. Each input has exactly one arrow leaving it if the mapping is a function. This visual tool is useful when the domain and range contain only a few elements.
映射图画出两个集合，并用箭头把每个输入连接到它的输出。如果这个映射是一个函数，那么每个输入恰好只有一条箭头离开。当定义域和值域元素不多时，这种可视化工具非常有用。
For example, if the domain is {1, 2, 3} and the rule is “double and add 1”, then 1 → 3, 2 → 5, 3 → 7. No input maps to two different outputs.
例如，若定义域为 {1, 2, 3}，规则是“乘以 2 再加 1”，则 1 → 3、2 → 5、3 → 7。没有输入会对应两个不同输出。
In set notation, we may define a function by a set of ordered pairs: {(1, 3), (2, 5), (3, 7)}. The first coordinate is the input, and the second coordinate is the output.
在集合记号中，我们可以用有序对集合来定义函数：{(1, 3), (2, 5), (3, 7)}。第一个坐标为输入，第二个坐标为输出。
8. Piecewise Functions | 分段函数
A piecewise function uses different rules for different parts of its domain. For example, the absolute value function can be written as |x| = x when x ≥ 0 and |x| = -x when x < 0.
分段函数在其定义域的不同区间使用不同的规则。例如，绝对值函数可以写成 |x| = x（当 x ≥ 0），|x| = -x（当 x < 0）。
When evaluating a piecewise function, first check which interval the input lies in, then apply the corresponding formula. For example, if g(x) = x² for x < 1 and g(x) = 2x for x ≥ 1, then g(0) = 0² = 0, but g(3) = 2(3) = 6.
求分段函数值时，先判断输入属于哪个区间，然后应用对应的公式。例如，若 g(x) = x²（x < 1），g(x) = 2x（x ≥ 1），则 g(0) = 0² = 0，但 g(3) = 2(3) = 6。
In A-Level, piecewise functions often appear in questions about continuity and integration. You must be careful at the boundary points and check whether the two rules agree there.
在 A-Level 中，分段函数经常出现在连续性和积分题目中。你必须特别注意分段点，并检查两段规则在该点是否一致。
9. Composite Functions | 复合函数
A composite function is formed when one function is applied after another. If f: x → 2x and g: x → x + 1, then fg(x) means “apply g first, then f”. The notation is fg(x) = f(g(x)).
复合函数是一个函数作用在另一个函数之后形成的。若 f: x → 2x，g: x → x + 1，那么 fg(x) 表示“先应用 g，再应用 f”。记号为 fg(x) = f(g(x))。
For example, g(3) = 4, then f(4) = 8, so fg(3) = 8. In general, fg(x) = 2(x + 1) = 2x + 2, while gf(x) = g(2x) = 2x + 1. This shows that fg(x) and gf(x) can be different.
例如，g(3) = 4，接着 f(4) = 8，所以 fg(3) = 8。一般情况下，fg(x) = 2(x + 1) = 2x + 2
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