Functions: Concepts and Common Exam Approaches | 函数的概念与常见考法

📚 Functions: Concepts and Common Exam Approaches | 函数的概念与常见考法

A function is one of the most fundamental concepts in A Level mathematics. It describes a relationship between two sets of values, where each input produces exactly one output. Mastering the definition, notation, and common exam question types is essential for success in any mathematics paper.

函数是 A Level 数学中最基本的概念之一。它描述两组值之间的关系，每个输入恰好产生一个输出。掌握函数的定义、记号以及常见考法，是在任何数学试卷中取得好成绩的关键前提。

1. Definition of a Function | 函数的定义

A function f from set A to set B is a rule that assigns to every element x in set A exactly one element y in set B. The key condition is uniqueness of output: for each input x, there is one and only one output f(x).

函数 f 从集合 A 到集合 B 是一种规则，它将集合 A 中的每一个元素 x 恰好对应到集合 B 中的一个元素 y。关键在于输出的唯一性：每个输入 x 有且仅有一个输出 f(x)。

f : x ↦ f(x), x ∈ A, f(x) ∈ B

For example, f(x) = x² is a function, because every real number x has exactly one square. However, the relation y² = x is not a function, because a single x-value (such as x = 4) corresponds to two y-values (y = ±2).

例如，f(x) = x² 是一个函数，因为每个实数 x 恰好有一个平方。然而，关系式 y² = x 不是函数，因为一个 x 值（如 x = 4）对应两个 y 值（y = ±2）。

2. Domain and Range | 定义域与值域

The domain of a function is the complete set of possible input values x. The range is the complete set of possible output values f(x). Correctly identifying the domain is often the first step in solving function problems.

函数的定义域是所有可能输入值 x 的集合。值域是所有可能输出值 f(x) 的集合。正确确定定义域通常是解决函数问题的第一步。

Common domain restrictions include: division by zero is not allowed (denominator ≠ 0); the expression under a square root must be ≥ 0; and logarithms require a positive argument. For example:

常见的定义域限制包括：分母不能为零；根号下的表达式必须 ≥ 0；对数函数的真数必须为正。例如：

For f(x) = 1/(x − 3), the domain is x ≠ 3.
对于 f(x) = 1/(x − 3)，定义域为 x ≠ 3。
For g(x) = √(x − 2), the domain is x ≥ 2, and the range is g(x) ≥ 0.
对于 g(x) = √(x − 2)，定义域为 x ≥ 2，值域为 g(x) ≥ 0。
For h(x) = ln(x), the domain is x > 0.
对于 h(x) = ln(x)，定义域为 x > 0。

3. Function Notation and Evaluation | 函数记号与求值

Function notation f(x) explicitly shows both the name of the function and the variable. Evaluating a function means substituting a given input value and simplifying. This is a direct, low-weight exam question that appears frequently.

函数记号 f(x) 同时明确地显示函数名称和变量。求函数值就是将给定的输入值代入并化简。这是一种直接、低难度的考试题目，出现频率很高。

Given f(x) = 2x² − 3x + 1, evaluate f(−2):

已知 f(x) = 2x² − 3x + 1，求 f(−2)：

f(−2) = 2(−2)² − 3(−2) + 1 = 8 + 6 + 1 = 15

Students should also be comfortable with composite expressions such as f(a + 1) or f(2x), where the substitution must be applied carefully to every occurrence of x.

学生还应熟练处理复合表达式，如 f(a + 1) 或 f(2x)，此时代入操作必须小心地应用于每一个 x。

4. Composite Functions | 复合函数

A composite function applies one function to the output of another. The notation fg(x) means f(g(x)): first apply g to x, then apply f to the result. The order matters — fg is generally not the same as gf.

复合函数是将一个函数作用于另一个函数的输出。记号 fg(x) 表示 f(g(x))：先将 g 作用于 x，再将 f 作用于结果。顺序非常重要——fg 通常不等于 gf。

fg(x) = f(g(x)), gf(x) = g(f(x))

For example, let f(x) = 2x + 1 and g(x) = x². Then fg(x) = f(x²) = 2x² + 1, while gf(x) = g(2x + 1) = (2x + 1)². Clearly, fg(x) ≠ gf(x).

例如，设 f(x) = 2x + 1，g(x) = x²。则 fg(x) = f(x²) = 2x² + 1，而 gf(x) = g(2x + 1) = (2x + 1)²。显然 fg(x) ≠ gf(x)。

When finding the domain of a composite function fg(x), you must ensure that x is in the domain of g, and that g(x) is in the domain of f.

求复合函数 fg(x) 的定义域时，必须确保 x 在 g 的定义域内，并且 g(x) 在 f 的定义域内。

5. Inverse Functions | 反函数

The inverse function f⁻¹ reverses the action of f. If f maps x to y, then f⁻¹ maps y back to x. A function has an inverse if and only if it is one-to-one (injective) on its domain.

反函数 f⁻¹ 逆转 f 的作用。如果 f 将 x 映射到 y，则 f⁻¹ 将 y 映射回 x。一个函数有反函数当且仅当它在定义域上是一一对应（单射）的。

The graph of f⁻¹ is the reflection of the graph of f in the line y = x. Key relationships are:

函数 f⁻¹ 的图像是 f 的图像关于直线 y = x 的反射。关键关系为：

ff⁻¹(x) = x and f⁻¹f(x) = x
ff⁻¹(x) = x 且 f⁻¹f(x) = x
Domain of f⁻¹ = Range of f; Range of f⁻¹ = Domain of f
f⁻¹ 的定义域 = f 的值域；f⁻¹ 的值域 = f 的定义域

To find f⁻¹ algebraically: write y = f(x), swap x and y, then solve for y. For instance, if f(x) = 3x − 2, then y = 3x − 2 → x = 3y − 2 → y = (x + 2)/3, so f⁻¹(x) = (x + 2)/3.

求 f⁻¹ 的代数方法：写出 y = f(x)，交换 x 和 y，然后解出 y。例如，若 f(x) = 3x − 2，则 y = 3x − 2 → x = 3y − 2 → y = (x + 2)/3，所以 f⁻¹(x) = (x + 2)/3。

6. One-to-One Functions and Restricting the Domain | 一一对应函数与限制定义域

Many functions, such as f(x) = x², are not one-to-one over their natural domain because both x and −x produce the same output. To define an inverse, we restrict the domain so that the function becomes one-to-one.

许多函数（如 f(x) = x²）在自然定义域上不是一一对应的，因为 x 和 −x 产生相同的输出。为了定义反函数，我们需要限制定义域，使函数成为一一对应。

For f(x) = x², restricting the domain to x ≥ 0 gives the inverse f⁻¹(x) = √x. Restricting to x ≤ 0 would instead give f⁻¹(x) = −√x. The examiner expects you to state your restricted domain clearly and to adjust the range of the inverse accordingly.

对于 f(x) = x²，将定义域限制为 x ≥ 0 可得到反函数 f⁻¹(x) = √x。限制为 x ≤ 0 则得到 f⁻¹(x) = −√x。考官期望你清楚地说明限制后的定义域，并相应调整反函数的值域。

7. Transforming Functions | 函数变换

Function transformations describe how the graph of y = f(x) changes under algebraic modifications. These are common exam questions, often requiring you to sketch the transformed graph or write down its equation.

函数变换描述 y = f(x) 的图像在代数修改下如何变化。这是常见的考试题目，通常要求你画出变换后的图像或写出其方程。

Transformation 变换	Effect 效果
y = f(x) + a	Vertical shift up by a (向上平移 a 个单位)
y = f(x + a)	Horizontal shift left by a (向左平移 a 个单位)
y = −f(x)	Reflection in the x-axis (关于 x 轴对称)
y = f(−x)	Reflection in the y-axis (关于 y 轴对称)
y = a·f(x)	Vertical stretch by factor a (纵向拉伸 a 倍)
y = f(ax)	Horizontal compression by factor 1/a (横向压缩为 1/a)

A common mistake is confusing y = f(x + a) with a shift to the right. Remember: inside the brackets, the shift direction is opposite to the sign.

一个常见错误是将 y = f(x + a) 误认为向右平移。记住：括号内的平移方向与符号相反。

8. Modulus Functions | 绝对值函数

The modulus function |x| gives the non-negative value of x: |x| = x when x ≥ 0, and |x| = −x when x < 0. Graphically, the modulus function reflects the negative portion of y = f(x) above the x-axis.

绝对值函数 |x| 给出 x 的非负值：当 x ≥ 0 时 |x| = x，当 x < 0 时 |x| = −x。从图像上看，绝对值函数将 y = f(x) 的负值部分沿 x 轴向上翻折。

Solving equations with modulus signs requires considering two cases. For example, to solve |2x − 1| = 5:

解含绝对值的方程需要考虑两种情况。例如，解 |2x − 1| = 5：

2x − 1 = 5 ⇒ x = 3
2x − 1 = −5 ⇒ x = −2

When solving inequalities such as |x − 2| < 3, the solution is −1 < x < 5, which can be found algebraically or by interpreting the modulus as a distance on the number line.

解不等式如 |x − 2| < 3 时，解为 −1 < x < 5，可以通过代数方法求解，也可以将绝对值理解为数轴上的距离。

9. Sketching Graphs of Functions | 绘制函数图像

Sketching is a core skill tested across all exam boards. A good sketch must show: the general shape of the curve, intercepts with the axes, turning points, and asymptotes where applicable. Labels are essential.

绘图是所有考试局都会考查的核心技能。一幅好的草图必须展示：曲线的大致形状、与坐标轴的交点、极值点以及渐近线（若存在）。标注是必不可少的。

For a quadratic f(x) = ax² + bx + c, determine the direction of the parabola from the sign of a, find the roots by factorising or using the quadratic formula, and locate the vertex using x = −b/(2a). For rational functions, identify vertical asymptotes where the denominator is zero and horizontal asymptotes from the degrees of numerator and denominator.

对于二次函数 f(x) = ax² + bx + c，通过 a 的符号判断抛物线开口方向，通过因式分解或求根公式找到根，并利用 x = −b/(2a) 确定顶点。对于有理函数，找出分母为零处的垂直渐近线，并根据分子分母的次数判断水平渐近线。

10. Common Exam Question Types | 常见考题类型

Exam questions on functions typically fall into a few recurring patterns. Recognising these patterns can significantly improve your speed and accuracy.

函数相关的考试题目通常可以归入几种重复出现的题型。识别这些题型可以显著提高你的速度和准确率。

Direct evaluation: substitute a value into a given function.
直接求值：将数值代入给定函数。
Composite functions: find fg(x) or a missing component given the composite result.
复合函数：求 fg(x)，或在已知复合结果时求缺失的函数分量。
Inverse functions: find f⁻¹(x) algebraically, often after restricting the domain.
反函数：通过代数方法求 f⁻¹(x)，通常先限制定义域。
Range problems: determine the range from the domain and the behaviour of the function.
值域问题：根据定义域和函数的行为确定值域。
Graph transformations: write the equation of a transformed graph or describe a transformation.
图像变换：写出变换后图像的方程或描述变换过程。
Modulus equations and inequalities: solve algebraically or by graph sketching.
绝对值方程与不等式：通过代数方法或图像法求解。

For questions involving the range, a useful strategy is to complete the square for quadratics. For f(x) = x² − 4x + 7 = (x − 2)² + 3, the minimum value is 3, so the range is f(x) ≥ 3.

对于涉及值域的题目，配方法对二次函数非常有用。例如 f(x) = x² − 4x + 7 = (x − 2)² + 3，最小值为 3，因此值域为 f(x) ≥ 3。

11. Common Mistakes to Avoid | 常见错误提醒

Students lose marks on function questions for a handful of repeated mistakes. Being aware of these pitfalls will help you avoid them in the exam.

学生在函数题上失分往往是因为几个重复出现的错误。了解这些陷阱将帮助你在考试中避免它们。

Forgetting domain restrictions when finding an inverse function.
求反函数时忘记注明定义域限制。
Swapping the order in composite functions: fg(x) ≠ gf(x) in general.
在复合函数中颠倒次序：一般地 fg(x) ≠ gf(x)。
Confusing horizontal shifts: y = f(x + 2) moves left, not right.
混淆水平平移方向：y = f(x + 2) 向左移动而非向右。
Forgetting that the modulus of a negative number is positive.
忘记负数的绝对值是正数。
Not checking whether a proposed inverse actually satisfies f(f⁻¹(x)) = x.
没有检验所求反函数是否真正满足 f(f⁻¹(x)) = x。

Always read the question carefully to see whether the domain has been specified. If it has, your range and any inverse must respect that domain.

务必仔细审题，看定义域是否已被限定。如果已限定，你的值域和反函数必须尊重该定义域。

12. Exam Strategy and Practice | 考试策略与练习建议

Function questions appear in nearly every A Level mathematics paper, often in both Section A (short questions) and Section B (extended problem-solving). Aim to secure the short marks first, then attempt the longer, multi-part questions that combine functions with other topics such as calculus or coordinate geometry.

函数题几乎出现在每一份 A Level 数学试卷中，通常既在 A 部分（短题）也在 B 部分（扩展综合题）出现。先稳拿短题的分数，再去尝试将函数与微积分或坐标几何等主题结合的长题。

When solving a multi-part function question, work systematically: part (a) often provides a result needed for part (b). If you cannot complete an earlier part, write down the method you would use — examiners award method marks generously.

解决多部分的函数题时，要系统性地推进：(a) 部分的结果通常服务于 (b) 部分。如果你无法完成前面的部分，写下你会使用的方法——考官在方法分上通常比较慷慨。

For revision, practise past-paper questions on functions from your specific board. Pay special attention to the mark schemes to understand exactly what notation and reasoning the examiner expects.

复习时，练习你所在考试局的函数历年真题。特别关注评分标准，以准确理解考官期望的记号和推理过程。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Functions: Concepts and Common Exam Approaches | 函数的概念与常见考法