A-Level Mathematics: Functions Concepts and Properties | A-Level 数学：函数概念与性质

📚 A-Level Mathematics: Functions Concepts and Properties | A-Level 数学：函数概念与性质

At its heart, a function is a precise rule that connects every allowed input to exactly one output. Understanding functions is the single most important skill in A-Level Mathematics: it underpins calculus, graphing, solving equations, and many real-world models.

函数的本质是一条精确的对应规则：每一个允许的输入值都对应唯一的一个输出值。在 A-Level 数学中，理解函数是最核心的基础能力，它贯穿微积分、绘制图像、解方程以及各类实际建模问题。

1. The Definition of a Function | 函数的定义

A function maps each element of its domain to exactly one element of its range. We often write this as f : x ↦ f(x). The symbol f names the rule, x is the input, and f(x) is the output.

函数将定义域中的每一个元素映射到值域中唯一的一个元素。我们常写成 f : x ↦ f(x)。其中 f 表示对应法则，x 是输入值，f(x) 是输出值。

For example, with f(x) = x² + 1, the input 2 is sent to 5 because 2² + 1 = 5. Every real number can be substituted into this rule, so the domain is all real numbers, x ∈ ℝ.

例如，对于 f(x) = x² + 1，输入值 2 会映射到 5，因为 2² + 1 = 5。因为任何实数都能代入该规则，所以定义域为所有实数，即 x ∈ ℝ。

f : x ↦ x² + 1, x ∈ ℝ

A key fact is that each input has only one output. If an input produced two different outputs, the relationship would not be a function.

关键事实是：每个输入只能有唯一输出。如果同一个输入对应两个不同的输出，那么这个关系就不是函数。

2. Domain and Range | 定义域与值域

The domain is the set of all allowed inputs. The range is the set of all possible outputs produced by those inputs. On a graph, the domain appears along the horizontal axis and the range along the vertical axis.

定义域是所有允许输入值构成的集合；值域是所有可能输出值构成的集合。在图像上，定义域沿水平方向展开，值域沿竖直方向展开。

Some functions have natural restrictions. For f(x) = √(x − 1) + 3, the expression inside the square root cannot be negative, so we need x − 1 ≥ 0, giving domain x ≥ 1. Since the square root is at least 0, the range is f(x) ≥ 3.

有些函数具有天然的限制。对于 f(x) = √(x−1) + 3，根号内的式子不能为负，因此需要 x−1 ≥ 0，即定义域为 x ≥ 1。由于根号的值至少为 0，所以值域为 f(x) ≥ 3。

f(x) = √(x − 1) + 3, x ≥ 1, range: f(x) ≥ 3

In CIE questions, you are often asked to state the domain and range before finding an inverse or sketching a graph. Always check for denominators, square roots, and logarithm restrictions.

在 CIE 考试中，常见题型是先写出定义域和值域，再求反函数或画出图像。解题时一定要检查分母是否为零、根号内是否非负、对数的真数是否大于零等限制条件。

3. One-to-One, Many-to-One, and One-to-Many Mappings | 单射、多对一与一对多映射

Linear functions such as f(x) = 3x + 1 are one-to-one: different inputs always give different outputs. Quadratic functions such as f(x) = x² are many-to-one when the domain is all real numbers because x and −x produce the same output.

线性函数如 f(x) = 3x + 1 是单射函数：不同的输入一定得到不同的输出。二次函数如 f(x) = x² 在定义域为全体实数时是多对一函数，因为 x 与 −x 会得到相同的输出。

A one-to-many mapping, such as y² = x, assigns more than one output to a single input. This is not a function. The vertical line test on a graph detects this: if a vertical line cuts the curve more than once, the curve is not a function.

一对多映射，比如 y² = x，会给同一个输入分配多个输出，因此它不是函数。图像上的竖直线检验可以判断这一点：如果一条竖直线与曲线相交多于一次，则该曲线不表示函数。

For a one-to-one function, every horizontal line will meet the graph at most once. This horizontal line test also tells us whether an inverse function exists.

对于单射函数，任何一条水平直线与图像最多相交一次。这个水平线检验也能告诉我们反函数是否存在。

4. Composite Functions | 复合函数

A composite function applies one function and then applies another. The notation fg(x) means “do g first, then do f”, so fg(x) = f(g(x)). The order matters: fg is usually different from gf.

复合函数是先执行一个函数，再执行另一个函数。记号 fg(x) 表示“先进行 g，再进行 f”，即 fg(x) = f(g(x))。运算顺序非常重要：fg 通常不等于 gf。

Let f(x) = 2x and g(x) = x² + 1. Then fg(x) = f(g(x)) = 2(x² + 1) = 2x² + 2, while gf(x) = g(f(x)) = (2x)² + 1 = 4x² + 1.

设 f(x) = 2x，g(x) = x² + 1。则 fg(x) = f(g(x)) = 2(x² + 1) = 2x² + 2；而 gf(x) = g(f(x)) = (2x)² + 1 = 4x² + 1。

fg(x) = 2x² + 2, gf(x) = 4x² + 1

When finding the domain of fg(x), you need x to be in the domain of g, and g(x) must also be in the domain of f. This two-part check is often missed by students.

求 fg(x) 的定义域时，既要保证 x 在 g 的定义域内，也要保证 g(x) 在 f 的定义域内。这个双重检查是很多学生容易遗漏的地方。

5. Inverse Functions | 反函数

An inverse function undoes the original function. If f(a) = b, then f⁻¹(b) = a. The notation f⁻¹(x) is read as “the inverse of f”. It is not the same as a reciprocal.

反函数用于“抵消”原函数的作用。如果 f(a) = b，那么 f⁻¹(b) = a。记号 f⁻¹(x) 读作“f 的反函数”，它不是倒数 1/f(x)。

For an inverse function to exist, the original function must be one-to-one over its given domain. Many-to-one functions do not have inverse functions unless we restrict their domain to make them one-to-one.

要使反函数存在，原函数在给定定义域上必须是单射函数。多对一函数没有反函数，除非我们限制其定义域，使它变成单射函数。

The domain of f⁻¹ is equal to the range of f, and the range of f⁻¹ is equal to the domain of f. The graphs of y = f(x) and y = f⁻¹(x) are reflections of each other in the line y = x.

f⁻¹ 的定义域等于 f 的值域，f⁻¹ 的值域等于 f 的定义域。函数 y = f(x) 与 y = f⁻¹(x) 的图像关于直线 y = x 对称。

6. Finding an Inverse Algebraically | 代数方法求反函数

To find the inverse of a one-to-one function, write y = f(x), then rearrange the equation to make x the subject. Finally, swap x and y and write the result as f⁻¹(x).

求一个单射函数的反函数时，先写出 y = f(x)，再通过代数变形把 x 表示为 y 的式子。最后交换 x 和 y，并把结果写成 f⁻¹(x)。

For example, let f(x) = 2x + 3, with domain x ∈ ℝ. Set y = 2x + 3. Subtract 3 and divide by 2 to get x = (y − 3)/2. Swap variables to obtain the inverse function.

例如，设 f(x) = 2x + 3，定义域为 x ∈ ℝ。令 y = 2x + 3，两边减 3 再除以 2，得到 x = (y − 3)/2。交换 x 与 y，就得到反函数。

f⁻¹(x) = (x − 3)/2

Check with a pair of values: f(4) = 11, and f⁻¹(11) = (11 − 3)/2 = 4. The composition f⁻¹(f(x)) should return x.

可以用一组数值检验：f(4) = 11，而 f⁻¹(11) = (11−3)/2 = 4。复合 f⁻¹(f(x)) 应当返回 x。

7. Graph Transformations | 图像变换

CIE Pure Mathematics questions often require you to sketch transformed graphs by starting with a known curve. Each transformation changes the coordinates in a predictable way.

CIE 纯数学题目经常要求你从已知曲线出发，画变换后的图像。每一种变换都会以可预测的方式改变坐标。

Function	Transformation
y = f(x) + a	Translation a units upward
y = f(x + a)	Translation a units left
y = −f(x)	Reflection in the x-axis
y = f(−x)	Reflection in the y-axis
y = k f(x)	Vertical stretch, scale factor k
y = f(kx)	Horizontal stretch, scale factor 1/k

If f(x) = x², then f(x) − 4 is x² − 4, a parabola shifted 4 units down. The graph of f(x − 2) = (x − 2)² is the same parabola shifted 2 units to the right.

若 f(x) = x²，则 f(x) − 4 就是 x² − 4，即抛物线向下平移 4 个单位。f(x − 2) = (x − 2)² 的图像则是原抛物线向右平移 2 个单位。

A common exam trap is translating in the horizontal direction. In y = f(x + a), the shift is a units in the negative x-direction, not the positive direction.

一个常见陷阱是水平方向平移。在 y = f(x + a) 中，平移方向是 x 轴的负方向，即向左平移 a 个单位，而不是向右。

8. The Modulus Function | 绝对值函数

The modulus function |x| gives the distance of x from zero, so it is always non-negative. Its piecewise definition is: |x| = x for x ≥ 0, and |x| = −x for x < 0.

绝对值函数 |x| 表示 x 到 0 的距离，因此结果永远非负。它的分段定义为：x ≥ 0 时 |x| = x；x < 0 时 |x| = −x。

|x| = x for x ≥ 0, |x| = −x for x < 0

The graph of y = |f(x)| is obtained by reflecting the part of y = f(x) that lies below the x-axis back above the x-axis. It always has a V-shape when f is linear.

y = |f(x)| 的图像可以通过将 y = f(x) 位于 x 轴下方的部分向上翻折得到。当 f 是线性函数时，图像呈 V 字形。

To solve |2x − 3| = 5, split into two cases: 2x − 3 = 5 or 2x − 3 = −5. This gives x = 4 or x = −1.

解方程 |2x − 3| = 5 时，需要分成两种情况：2x − 3 = 5 或 2x − 3 = −5。解得 x = 4 或 x = −1。

9. Even and Odd Functions | 偶函数与奇函数

An even function satisfies f(−x) = f(x) for every x in its domain. Its graph is symmetrical about the y-axis. Examples include f(x) = x² and f(x) = cos x.

偶函数满足对定义域内每一个 x 都有 f(−x) = f(x)。它的图像关于 y 轴对称。典型例子包括 f(x) = x² 和 f(x) = cos x。

An odd function satisfies f(−x) = −f(x) for every x in its domain. Its graph has rotational symmetry of order 2 about the origin. Examples include f(x) = x³ and f(x) = sin x.

奇函数满足对定义域内每一个 x 都有 f(−x) = −f(x)。它的图像关于原点旋转 180° 后与原图重合。典型例子包括 f(x) = x³ 和 f(x) = sin x。

To test a function algebraically, substitute −x into f(x) and simplify. If the result is the original expression, it is even; if it is the negative of the original expression, it is odd. Otherwise, it is neither.

检验时，将 −x 代入 f(x) 并化简。如果结果等于原表达式，则为偶函数；如果结果等于原表达式的相反数，则为奇函数；否则既不是奇函数也不是偶函数。

10. Increasing and Decreasing Functions | 递增函数与递减函数

A function is increasing on an interval if f(x₂) ≥ f(x₁) whenever x₂ > x₁. It is decreasing if f(x₂) ≤ f(x₁) whenever x₂ > x₁. For differentiable functions, this is related to the sign of the derivative.

如果对于 x₂ > x₁ 总有 f(x₂) ≥ f(x₁)，则函数在该区间上递增；如果总有 f(x₂) ≤ f(x₁)，则函数在该区间上递减。对于可导函数，这可以由导数的符号判断。

If f'(x) > 0 on an interval, the function is increasing there. If f'(x) < 0, the function is decreasing. At a stationary point, f'(x) = 0, and the function may have a local maximum, local minimum, or a point of inflection.

若在某区间上 f'(x) > 0，则函数递增；若 f'(x) < 0，则函数递减。在驻点处 f'(x) = 0，此时函数可能有局部最大值、局部最小值或拐点。

Consider f(x) = x² − 4x. Its derivative is f'(x) = 2x − 4. Setting 2x − 4 = 0 gives x = 2. Since the graph is a parabola opening upward, x = 2 is the location of the minimum point.

考虑 f(x) = x² − 4x。其导数为 f'(x) = 2x − 4。令 2x − 4 = 0，得 x = 2。由于图像是开口向上的抛物线，因此 x = 2 对应最小值点。

f(x) = x² − 4x, minimum at x = 2

11. Domain Restrictions and Inverse Functions | 定义域限制与反函数

When a many-to-one function has a restricted domain, it can become one-to-one. For example, f(x) = x² with domain x ≥ 0 is one-to-one, so its inverse exists.

当一个多对一函数被限定定义域后，可能变成单射函数。例如 f(x) = x²，在定义域 x ≥ 0 上就是单射函数，因此它存在反函数。

For f(x) = x², x ≥ 0, the range is f(x) ≥ 0, and the inverse is f⁻¹(x) = √x, x ≥ 0. If instead the domain were x ≤ 0, the inverse would be f⁻¹(x) = −√x.

对于 f(x) = x² 且 x ≥ 0，值域为 f(x) ≥ 0，反函数为 f⁻¹(x) = √x，x ≥ 0。如果把定义域限制为 x ≤ 0，则反函数应为 f⁻¹(x) = −√x。

f(x) = x², x ≥ 0 ⇒ f⁻¹(x) = √x

This idea is frequently tested in CIE questions about composite and inverse functions. A correct domain statement is essential for a correct inverse function.

这个思想在 CIE 考试中经常与复合函数、反函数结合考查。正确的定义域表述是求得正确反函数的关键。

12. Problem-Solving Strategy | 解题策略

Start by identifying the type of function and its natural domain. Sketch a graph if possible, then state the domain and range carefully before attempting composition, inversion, or transformation.

解题时，先判断函数类型并确定自然定义域。如果可能，先画出函数图像，再仔细写出定义域和值域，然后进行复合、求反函数或图像变换。

When solving functional equations, always check whether your answer lies in the domain. With modulus equations, remember to test both cases. With composite functions, keep track of which rule is applied first.

在解函数方程时，务必检验答案是否在定义域内。遇到绝对值方程，要记得讨论两种情形。遇到复合函数，要分清哪个函数先执行。

Functions are more than abstract rules; they describe real quantities such as cost, population, displacement, and voltage. Mastering function notation early will make every later topic in A-Level Mathematics easier.

函数不仅仅是抽象的规则，它可以描述成本、人口、位移和电压等实际量。尽早掌握函数记号，会让 A-Level 数学中后续所有专题都变得更加轻松。

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A-Level Mathematics: Functions Concepts and Properties | A-Level 数学：函数概念与性质