Functions and Mappings: Core Concepts | 函数与映射的核心概念

📚 Functions and Mappings: Core Concepts | 函数与映射的核心概念

A function is one of the most fundamental ideas in A-Level Mathematics. It describes a precise relationship between two sets, where each input gives exactly one output. Understanding functions and mappings is essential for calculus, algebra, and modelling real-world problems.

函数是 A-Level 数学中最基本的概念之一。它描述了两个集合之间一种精确的关系：每一个输入都给出唯一的输出。理解函数与映射是学习微积分、代数以及解决实际问题的关键基础。

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 each element \( x \) in A exactly one element \( y \) in B. We write \( f : A \rightarrow B \) and \( y = f(x) \). The key condition is uniqueness: one input cannot produce two different outputs.

函数 \( f \) 是从集合 A 到集合 B 的一种规则，它将 A 中的每一个元素 \( x \) 都唯一地对应到 B 中的一个元素 \( y \)。我们记作 \( f : A \rightarrow B \) 且 \( y = f(x) \)。关键条件是唯一性：一个输入不能产生两个不同的输出。

For example, \( f(x) = x^2 \) is a function because every real number \( x \) gives exactly one squared value. However, the relation \( y^2 = x \) is not a function of \( x \), because a positive \( x \) gives two possible \( y \) values: \( +\sqrt{x} \) and \( -\sqrt{x} \).

例如，\( f(x) = x^2 \) 是一个函数，因为每个实数 \( x \) 都对应唯一一个平方值。然而，关系式 \( y^2 = x \) 不是关于 \( x \) 的函数，因为正数 \( x \) 会对应两个可能的 \( y \) 值：\( +\sqrt{x} \) 和 \( -\sqrt{x} \)。

2. Types of Mappings | 映射的类型

Mappings can be classified into four main types: one-to-one, many-to-one, one-to-many, and many-to-many. In A-Level mathematics, we usually focus on the first three, and only functions are allowed to be one-to-one or many-to-one.

映射可以分为四类：一一映射、多一映射、一多映射和多多映射。在 A-Level 数学中，我们通常关注前三类，而且只有一一映射和多一映射才属于函数。

One-to-one mapping: each input has a distinct output, and each output comes from exactly one input. Example: \( f(x) = 2x + 1 \).
一一映射：每个输入都有唯一的输出，且每个输出恰好来自一个输入。例如：\( f(x) = 2x + 1 \)。

Many-to-one mapping: two or more inputs share the same output. Example: \( f(x) = x^2 \), where \( f(-2) = f(2) = 4 \).
多一映射：两个或多个输入对应同一个输出。例如：\( f(x) = x^2 \)，其中 \( f(-2) = f(2) = 4 \)。

One-to-many relation: one input gives several outputs. This is not a function. Example: \( y = \pm \sqrt{x} \).
一多关系：一个输入对应多个输出。这不是函数。例如：\( y = \pm \sqrt{x} \)。

3. Domain, Range and Codomain | 定义域、值域与余域

The domain is the set of all possible input values. The codomain is the set that contains all possible output values, while 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。余域是包含所有可能输出值的集合，而值域是函数实际产生的输出值所构成的余域的子集。

For \( f(x) = \sqrt{x-3} \), the domain is \( x \geq 3 \), because the expression under the square root cannot be negative. The range is \( f(x) \geq 0 \), because a square root is never negative.

对于 \( f(x) = \sqrt{x-3} \)，定义域是 \( x \geq 3 \)，因为根号下的表达式不能为负。值域是 \( f(x) \geq 0 \)，因为平方根永远不会是负数。

When finding the domain, look for restrictions such as denominators not equal to zero, even-index roots requiring non-negative radicands, and logarithms requiring positive arguments.

在寻找定义域时，要注意限制条件：分母不能为零，偶次根号下的数必须非负，对数函数中的真数必须为正。

4. Composite Functions | 复合函数

Given two functions \( f(x) \) and \( g(x) \), the composite function \( fg(x) \) means “apply \( g \) first, then apply \( f \) to the result.” We write \( fg(x) = f(g(x)) \). The order matters: \( fg(x) \) is generally not equal to \( gf(x) \).

给定两个函数 \( f(x) \) 和 \( g(x) \)，复合函数 \( fg(x) \) 表示“先运用 \( g \)，再把结果代入 \( f \)”。我们写作 \( fg(x) = f(g(x)) \)。顺序很重要：\( fg(x) \) 通常不等于 \( gf(x) \)。

Example: If \( f(x) = 2x \) and \( g(x) = x^2 + 1 \), then \( fg(x) = 2(x^2 + 1) = 2x^2 + 2 \), while \( gf(x) = (2x)^2 + 1 = 4x^2 + 1 \).

For composite functions, the domain is restricted to values of \( x \) for which \( g(x) \) lies in the domain of \( f \). Check whether the inner function produces valid inputs for the outer function.

对于复合函数，其定义域限于那些使 \( g(x) \) 落在 \( f \) 的定义域内的 \( x \) 取值。要检验内层函数的输出是否可以作为外层函数的输入。

5. Inverse Functions | 反函数

The inverse of a function \( f \), written \( f^{-1}(x) \), “undoes” the effect of \( f \). If \( y = f(x) \), then \( x = f^{-1}(y) \). The domain of \( f^{-1} \) is the range of \( f \), and the range of \( f^{-1} \) is the domain of \( f \).

函数 \( f \) 的反函数记为 \( f^{-1}(x) \)，它“抵消” \( f \) 的作用。如果 \( y = f(x) \)，那么 \( x = f^{-1}(y) \)。\( f^{-1} \) 的定义域是 \( f \) 的值域，而 \( f^{-1} \) 的值域是 \( f \) 的定义域。

To find the inverse of a one-to-one function: replace \( f(x) \) with \( y \), swap \( x \) and \( y \), then solve for \( y \). For example, \( f(x) = 3x + 2 \) becomes \( x = 3y + 2 \), so \( y = \frac{x – 2}{3} \). Thus \( f^{-1}(x) = \frac{x – 2}{3} \).

求一个一一函数的反函数步骤：将 \( f(x) \) 替换为 \( y \)，交换 \( x \) 和 \( y \)，然后解出 \( y \)。例如 \( f(x) = 3x + 2 \) 变为 \( x = 3y + 2 \)，所以 \( y = \frac{x – 2}{3} \)。因此 \( f^{-1}(x) = \frac{x – 2}{3} \)。

Graphically, the inverse function is the reflection of the original function in the line \( y = x \). If a function is not one-to-one, it does not have an inverse over its whole domain unless we restrict the domain.

在图像上，反函数是原函数关于直线 \( y = x \) 的镜像。如果一个函数不是一一函数，那么在整个定义域上它就没有反函数，除非我们限制其定义域。

6. The Horizontal Line Test and One-to-One Functions | 水平线检验与一一映射

A function is one-to-one if and only if every horizontal line intersects its graph at most once. This is called the horizontal line test. If a horizontal line crosses the graph more than once, the function has multiple inputs giving the same output.

一个函数是一一函数，当且仅当每一条水平直线与该函数图像的交点至多只有一个。这称为水平线检验。如果一条水平直线与图像相交超过一次，说明有多个输入对应同一个输出。

For example, \( f(x) = x^2 \) fails the horizontal line test because the line \( y = 4 \) meets the graph at \( x = -2 \) and \( x = 2 \). Therefore \( x^2 \) is not one-to-one on its natural domain. However, if we restrict the domain to \( x \geq 0 \), then \( f(x) = x^2 \) becomes one-to-one and its inverse is \( f^{-1}(x) = \sqrt{x} \).

例如，\( f(x) = x^2 \) 不能通过水平线检验，因为直线 \( y = 4 \) 与图像在 \( x = -2 \) 和 \( x = 2 \) 处相交。因此 \( x^2 \) 在其自然定义域上不是一一函数。但如果将定义域限制为 \( x \geq 0 \)，那么 \( f(x) = x^2 \) 就变成一一函数，其反函数为 \( f^{-1}(x) = \sqrt{x} \)。

7. Odd and Even Functions | 奇函数与偶函数

A function is even if \( f(-x) = f(x) \) for all \( x \) in its domain. Even functions are symmetric about the \( y \)-axis. Common examples include \( f(x) = x^2 \) and \( f(x) = \cos x \).

如果对于定义域中的所有 \( x \)，都有 \( f(-x) = f(x) \)，则称该函数为偶函数。偶函数关于 \( y \)-轴对称。常见的例子包括 \( f(x) = x^2 \) 和 \( f(x) = \cos x \)。

A function is odd if \( f(-x) = -f(x) \) for all \( x \) in its domain. Odd functions have rotational symmetry of 180° about the origin. Examples include \( f(x) = x^3 \) and \( f(x) = \sin x \).

如果对于定义域中的所有 \( x \)，都有 \( f(-x) = -f(x) \)，则称该函数为奇函数。奇函数关于原点具有 180° 旋转对称性。例子包括 \( f(x) = x^3 \) 和 \( f(x) = \sin x \)。

Many functions are neither odd nor even, such as \( f(x) = x^2 + x \). To test, compute \( f(-x) \) and compare it with \( f(x) \) and \( -f(x) \).

许多函数既不是奇函数也不是偶函数，例如 \( f(x) = x^2 + x \)。检验方法是计算 \( f(-x) \)，然后分别与 \( f(x) \) 和 \( -f(x) \) 比较。

8. Transformations of Functions | 函数的变换

Understanding how transformations affect graphs is crucial. For a function \( y = f(x) \):

理解变换如何影响图像至关重要。对于函数 \( y = f(x) \)：

Transformation	Change to graph	变换	图像变化
\( f(x) + a \)	Vertical translation by \( a \)	向上平移 \( a \) 个单位（\( a>0 \)）	纵向平移
\( f(x + a) \)	Horizontal translation by \( -a \)	向左平移 \( a \) 个单位（\( a>0 \)）	横向平移
\( -f(x) \)	Reflection in the \( x \)-axis	关于 \( x \)-轴对称	反射
\( f(-x) \)	Reflection in the \( y \)-axis	关于 \( y \)-轴对称	反射
\( af(x) \)	Vertical stretch by factor \( a \)	纵向拉伸，倍数为 \( a \)	伸缩
\( f(ax) \)	Horizontal stretch by factor \( \frac{1}{a} \)	横向压缩，倍数为 \( \frac{1}{a} \)	伸缩

Be careful with horizontal transformations: they work opposite to intuition. For example, \( f(x + 2) \) shifts the graph to the left by 2 units, not to the right.

要特别注意横向变换：它们与直觉相反。例如，\( f(x + 2) \) 将图像向左平移 2 个单位，而不是向右。

9. Finding the Range from a Graph | 从图像求值域

The range of a function can often be read directly from its graph. Look at the lowest and highest \( y \)-values that the graph reaches, and note any gaps or asymptotes.

函数的值域通常可以直接从图像中读出。观察图像能达到的最低和最高 \( y \)-值，并注意任何间断点或渐近线。

For example, the graph of \( f(x) = e^x \) approaches \( 0 \) as \( x \rightarrow -\infty \) but never reaches it, and grows without bound as \( x \rightarrow \infty \). Hence the range is \( y > 0 \).

例如，函数 \( f(x) = e^x \) 的图像在 \( x \rightarrow -\infty \) 时趋近于 0 但永远达不到，而在 \( x \rightarrow \infty \) 时无限增大。因此值域为 \( y > 0 \)。

For rational functions, check for horizontal and vertical asymptotes. The range excludes values that the function approaches but never equals, unless a point is specifically defined.

对于有理函数，要检查水平和垂直渐近线。值域要排除那些函数趋近但永远不能相等的值，除非某个点被特别定义。

10. Common Exam Pointers and Pitfalls | 常见考点与易错点

In exams, students often lose marks by confusing domain and range, forgetting domain restrictions when finding inverses, or applying transformations in the wrong order. Always state the domain and range clearly when asked.

在考试中，学生常常因为混淆定义域和值域、求反函数时忘记限制定义域，或变换顺序错误而失分。当题目要求时，务必清晰地写出定义域和值域。

Another common mistake is assuming that \( fg(x) = gf(x) \). Composite functions are not commutative in general. Always check whether the required inverse function exists before finding it.

另一个常见错误是认为 \( fg(x) = gf(x) \)。复合函数通常不满足交换律。在求反函数之前，一定要先检查反函数是否存在。

Final tip: practice sketching graphs quickly, because many questions about functions can be solved by visual reasoning. Use test points to confirm behaviour near asymptotes and critical points.

最后的建议：练习快速绘制图像，因为许多关于函数的问题可以通过图形推理来解决。使用测试点来确认渐近线和关键点附近的行为。

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Functions and Mappings: Core Concepts | 函数与映射的核心概念