📚 Functions and Graphs – Edexcel A-Level Pure Mathematics | 函数与图像 – 爱德思 A-Level 纯数
Understanding functions is at the heart of A-Level Mathematics, and the Edexcel specification dedicates significant attention to the language of functions, their graphs, and transformations. This article consolidates the key ideas from the core pure content – often found in Chapter 4 of many textbooks – covering domain, range, composite and inverse functions, graph transformations, and the modulus function. Whether you are preparing for AS or the full A-Level, mastering these concepts will sharpen your algebraic skills and deepen your graphical intuition.
理解函数是 A-Level 数学的核心，爱德思考试大纲对函数的语言、图像和变换给予了高度重视。本文整合了纯数核心内容中的关键思想——常见于许多教材第 4 章——涵盖定义域、值域、复合函数与反函数、图像变换以及绝对值函数。无论你是在备考 AS 还是完整 A-Level，掌握这些概念都将提升你的代数能力并加深你对图像的理解。
1. The Language of Functions | 函数的语言
A function is a rule that assigns each input exactly one output. We often write f(x) to represent the output when x is the input. For a mapping to be a function, every element of the domain must map to one and only one element of the codomain. In Edexcel exams, you may be asked to distinguish between a function and a general relation by examining ordered pairs or graphs.
函数是一种规则，为每个输入指定唯一的输出。我们常用 f(x) 表示在以 x 为输入时的输出。要使一个映射成为函数，定义域中的每一个元素都必须映射到值域中的唯一一个元素。在爱德思考试中，你可能会被要求通过检查有序对或图像来区分函数与一般关系。
A graph represents a function if any vertical line cuts it at most once. This is the vertical line test. However, a circle fails this test, so x² + y² = r² does not define y as a function of x.
如果任意一条竖直线与图像最多只有一个交点，则该图像表示一个函数。这就是铅垂线检验。然而，圆不满足该检验，因此 x² + y² = r² 并不能将 y 定义为 x 的函数。
2. Domain and Range | 定义域与值域
The domain of a function is the set of all possible input values (x), while the range is the set of all possible output values (f(x)). In A-Level problems, you will often need to determine the maximal domain for a given formula or restrict the domain to make the inverse exist.
函数的定义域是所有可能输入值 (x) 的集合，而值域是所有可能输出值 (f(x)) 的集合。在 A-Level 题目中，你经常需要确定给定表达式的最大定义域，或限制定义域以使反函数存在。
ional functions, the denominator cannot be zero. For example, the maximal domain of f(x) = √x − 2 te domains and ranges using set notation, interval notation, or inequalities as specified.
对于平方根函数，根号内的表达式必须 ≥ 0。对于有理函数，分母不能为零。例如，f(x) = √x − 2 的最大定义域是 x ≥ 2。像 f(x) = x² + 1 这样的二次函数的值域是 f(x) ≥ 1。要根据题目要求使用集合符号、区间符号或不等式来表示定义域和值域。
3. Composite Functions | 复合函数
A composite function is formed when one function is applied after another. We write fg(x) to mean f(g(x)), applying g first, then f. The domain of the composite function fg is restricted to those x in the domain of g for which g(x) lies in the domain of f.
复合函数是通过将一个函数作用于另一个函数的结果而形成的。我们用 fg(x) 表示 f(g(x))，即先应用 g，再应用 f。复合函数 fg 的定义域被限制为那些在 g 的定义域内且 g(x) 落在 f 的定义域中的 x。
xample, if f(x) = 2x + 3 and g(x) = x², then fg(x) = 2x² + 3. Graphically, a composite transformation corresponds to a sequence of elementary graph shifts.
在爱德思的题目中，你可能需要求出 fg(x) 的表达式，并说明其定义域与值域。务必仔细代换并化简。例如，若 f(x) = 2x + 3，g(x) = x²，则 fg(x) = 2x² + 3。从图像上看，复合变换对应于一系列基本图像移动。
4. Inverse Functions | 反函数
x), swap x and y, then solve for y. The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.
反函数 f⁻¹(x) 会逆转 f 的作用。只有当 f 在其定义域上是一一映射时，反函数才存在。从代数上讲，求反函数的方法是：写出 y = f(x)，交换 x 与 y，然后解出 y。f⁻¹ 的定义域就是 f 的值域，f⁻¹ 的值域就是 f 的定义域。
y = x. This symmetry is a common visual check in exams. When a domain restriction is given—for example, making a quadratic one-to-one—always state the restricted domain clearly.
y = f(x) 与 y = f⁻¹(x) 的图像关于直线 y = x 对称。这种对称性是考试中常见的图像验查方法。当题目给出了定义域的限制——例如，为使二次函数成为一一映射——务必清晰地写出这个受限制的定义域。
5. Graph Transformations: Translations | 图像变换：平移
Translations shift the whole graph without changing its shape. The transformation f(x) + a moves the graph vertically: upwards if a > 0, downwards if a < 0. The transformation f(x + a) moves the graph horizontally: left if a > 0, right if a < 0. Remember the horizontal shift is in the opposite direction to the sign of a.
平移移动整个图像而不改变其形状。变换 f(x) + a 使图像垂直移动：若 a > 0 则向上，若 a < 0 则向下。变换 f(x + a) 使图像水平移动：若 a > 0 则向左，若 a < 0 则向右。请记住，水平移动的方向与 a 的符号相反。
ypical exam question might ask you to sketch y = f(x + 2) − 3 given the graph of y = f(x).
用向量符号表示，列向量 [p, q] 所代表的平移将 f(x) 映射为 f(x − p) + q。典型的考试题目可能会给出 y = f(x) 的图像，然后要求你画出 y = f(x + 2) − 3 的草图。
6. Graph Transformations: Stretches and Reflections | 图像变换：伸缩与反射
a f(x) is a vertical stretch by factor a. If a > 1 the graph is stretched away from the x‑ ompressed. y = f(ax) is a horizontal stretch by factor 1/a. A stretch factor of 2 in the x‑direction corresponds to a = ½ inside the function.
伸缩会缩放图像。y = a f(x) 是以因子 a 进行的垂直伸缩。若 a > 1，图像背离 x 轴拉伸；若 0 < a < 1，则被压缩。y = f(ax) 是以因子 1/a 进行的水平伸缩。x 方向上的伸长因子 2 对应函数内部的 a = ½。
Reflections flip the graph. y = −f(x) reflects in the x‑axis; y = f(−x) reflects in the y‑ are present, apply stretches first, then translations, unless brackets indicate otherwise.
反射会翻转图像。y = −f(x) 关于 x 轴反射；y = f(−x) 关于 y 轴反射。这些变换可以组合使用，但应用顺序很重要。当同时存在伸缩与平移时，除非括号另有说明，否则先进行伸缩再进行平移。
7. The Modulus Function | 绝对值函数
The modulus function is defined by |x| = x when x ≥ 0, and |x| = −x when x < 0. It always outputs a non-negative value. Graphically, y = |f(x)| takes any portion of f(x) that lies below the x‑axis and reflects it upwards. The graph never goes below the x‑axis.
绝对值函数的定义为：当 x ≥ 0 时 |x| = x，当 x < 0 时 |x| = −x。它总是输出非负值。从图像上看，y = |f(x)| 会将 f(x) 在 x 轴下方的部分向上反射。图像绝不会低于 x 轴。
For y = f(|x|), the graph retains the part for x ≥ 0 and then reflects it in the y‑axis for negative x. This produces a graph that is symmetric about the y‑ tions often ask you to sketch graphs of modulus functions and to interpret the resulting V‑shapes.
对于 y = f(|x|)，图像会保留 x ≥ 0 的部分，然后将其关于 y 轴反射以得到负 x 区域的图像。这样得到的图像关于 y 轴对称。爱德思试题通常会要求你画出绝对值函数的草图并解释其 V 形特征。
8. Solving Modulus Equations and Inequalities | 解含绝对值的方程与不等式
An equation like |2x − 1| lways check that solutions satisfy the original modulus equation. For inequalities such as |x + 3| < 2, rewrite as −2 < x + 3 < 2 and solve to find the interval.
像 |2x − 1| = 5 这样的方程会转化为两个线性方程：2x − 1 = 5 或 2x − 1 = −5。务必检查解是否满足原绝对值方程。对于像 |x + 3| < 2 这样的不等式，可将其改写为 −2 < x + 3 < 2，然后求解得出区间。
For more complicated inequalities, e.g., |x − 1| > 2|x + 2|, a common method is to square both sides (since both sides are non‑negative) or to consider critical values and test regions. Sketching the graphs can also help visualise the solution set.
对于更复杂的不等式，如 |x − 1| > 2|x + 2|，常用方法是将两边平方（因为两边均非负），或者考虑临界值并对区间进行检验。画出函数图像也有助于直观看出解集。
9. Combining Transformations | 组合变换
. As a general rule, apply horizontal transformations in a way that directly affects the x‑ ertically by factor 2, and finally translate down by 1. However, if the function is written as y = 2 f(3x − 1), factorise the inside: f(3(x − ⅓)), revealing a horizontal translation of ⅓ to the right followed by a horizontal stretch with factor ⅓ (i.e. compression).
当函数经历多重变换时，必须小心处理变换顺序。一般来说，水平变换应以直接作用于 x 变量的方式进行，而垂直变换则作用于整个函数。例如，要画出 y = 2 f(x + 3) − 1 的图像，应先向左平移 3，然后垂直伸长因子 2，最后向下平移 1。然而，如果函数写作 y = 2 f(3x − 1)，则要对内部进行因式分解：f(3(x − ⅓))，这表明先向右平移 ⅓，再以因子 ⅓ 进行水平伸长（即压缩）。
ions are: horizontal translation by c, horizontal stretch by 1/b, vertical stretch by A, and vertical translation by d. Mastering this format will allow you to describe and sketch combined transformations accurately.
始终将函数看作 y = A f(b(x − c)) + d 的形式。其变换依次为：水平平移 c，水平伸长 1/b，垂直伸长 A，以及垂直平移 d。掌握这一格式将使你能够准确地描述并画出组合变换的图像。
10. Key Tips for Exam Success | 考试成功的关键提示
Read domain restrictions carefully; they often determine whether an inverse exists. When finding range, sketch a quick graph or analyse the function's behaviour at critical points and asymptotes. For composite functions, check the domain of the inside function first and then ensure the output fits the domain of the outer function. In modulus questions, always consider both the algebraic split method and the graphical approach to verify your answers. Finally, when giving final answers for transformations, use precise language such as 'translation by vector (−2, 5)' rather than vague descriptions.
仔细阅读定义域的限制条件；它们往往决定了反函数是否存在。在求值域时，快速画出草图或分析函数在临界点和渐近线附近的性态。对于复合函数，先检查内部函数的定义域，并确保其输出适合外部函数的定义域。在绝对值问题中，务必同时考虑代数分段方法和图像方法以验证答案。最后，在给出关于变换的最终答案时，请使用精确的语言，例如“平移向量为 (−2, 5)”，而非含糊的描述。
Practice with past papers from Edexcel Pure Mathematics, focusing on questions that link multiple topics—for example, combining inverse functions with graph transformations or modulus inequalities with quadratic curves. Being fluent in function notation and transformation order will save you valuable time in the exam.
使用爱德思纯数的历年试卷进行练习，重点关注那些将多个主题联系在一起的题目——例如，反函数与图像变换的结合，或绝对值不等式与二次曲线的结合。熟练掌握函数符号与变换顺序将为你在考试中赢得宝贵的时间。
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