Relations and Functions | 关系与函数

📚 Relations and Functions | 关系与函数

Understanding relations and functions is fundamental to higher-level mathematics in the IB programme. A relation pairs elements from one set to another, while a function is a special type of relation where each input is associated with exactly one output. This article explores the core concepts, notation, types, operations, and graphical interpretations necessary for mastery in IB Mathematics. We will cover domain and range, composite and inverse functions, transformations, and more.

理解关系与函数是 IB 数学高阶学习的基础。关系将一个集合中的元素与另一个集合中的元素配对，而函数是一种特殊的关系，其中每个输入恰好与一个输出相关联。本文探讨掌握 IB 数学所必需的核心概念、符号、类型、运算以及图像解释。我们将涵盖定义域与值域、复合函数与反函数、变换等内容。

1. What is a Relation? | 什么是关系？

A relation is any set of ordered pairs (x, y) that connects elements from a first set (the domain) to elements of a second set (the range). Relations can be represented as sets, mapping diagrams, tables, equations, or graphs. For example, {(1,2), (2,4), (3,6)} is a relation where the y-value is twice the x-value.

关系是任何一组有序对 (x, y)，它将第一个集合（定义域）的元素与第二个集合（值域）的元素连接起来。关系可以表示为集合、映射图、表格、方程或图像。例如，{(1,2), (2,4), (3,6)} 就是一个关系，其中 y 值是 x 值的两倍。

A relation does not require each x to have a unique y; a single input can be linked to multiple outputs. For instance, the circle equation x² + y² = 1 defines a relation where x = 0 connects to both y = 1 and y = -1, which is not a function.

关系不要求每个 x 都有唯一的 y；一个输入可以连接到多个输出。例如，圆方程 x² + y² = 1 定义的关系中，x = 0 同时对应 y = 1 和 y = -1，这就不是函数。

2. Defining a Function | 函数的定义

A function is a relation in which every element of the domain is paired with exactly one element of the codomain. We often write f : A → B, meaning f maps set A to set B. The defining property is: if (a, b) and (a, c) belong to f, then b = c.

函数是一种关系，其中定义域中的每个元素都恰好与陪域中的一个元素配对。我们通常写作 f : A → B，表示 f 将集合 A 映射到集合 B。其定义性质是：如果 (a, b) 和 (a, c) 都属于 f，则 b = c。

In IB exams, identifying functions from graphs often relies on the vertical line test: if any vertical line intersects the graph at more than one point, the graph does not represent a function. Functions are typically named with letters like f, g, h, and we express the rule as f(x) = …

在 IB 考试中，通过图像识别函数通常依赖于垂直线检验：如果任意一条垂直线与图像相交于多于一个点，则该图像不代表函数。函数通常用字母 f、g、h 命名，我们将规则表示为 f(x) = …

3. Domain, Codomain, and Range | 定义域、陪域与值域

The domain of a function is the set of all possible input values (x-values) for which the function is defined. The codomain is the set into which the outputs are supposed to fall, while the range (or image) is the set of actual output values produced by the function.

函数的定义域是所有可能输入值（x 值）的集合，对于这些值函数有定义。陪域是输出值应落入的集合，而值域（或像集）是函数实际产生的输出值的集合。

For a function f(x) = √(x – 2), the domain is x ≥ 2, the codomain might be ℝ, but the range is y ≥ 0. IB questions often ask students to determine the largest possible domain for a given formula, considering restrictions like division by zero or even roots of negative numbers.

对于函数 f(x) = √(x – 2)，定义域是 x ≥ 2，陪域可能是 ℝ，但值域是 y ≥ 0。IB 试题常要求学生确定给定公式的最大可能定义域，需考虑分母为零或负数的偶次方根等限制。

4. Function Notation and Evaluation | 函数符号与求值

Function notation uses f(x) to denote the output when x is the input. To evaluate f(3) for f(x) = 2x + 1, we substitute 3 into the rule to get 7. This notation also allows us to express combinations like f(a + h) and simplify expressions such as the difference quotient.

函数符号使用 f(x) 来表示当输入为 x 时的输出。对于 f(x) = 2x + 1，要计算 f(3)，我们将 3 代入规则得到 7。这种符号还允许我们表达诸如 f(a + h) 的组合，并简化差商等表达式。

Piecewise-defined functions use different rules for different intervals of the domain. For example, f(x) = { x², x < 0; 2x, x ≥ 0 } requires evaluating the input based on which condition it satisfies. IB problems often include graphing such piecewise functions.

分段定义的函数对定义域的不同区间使用不同的规则。例如 f(x) = { x², x < 0; 2x, x ≥ 0 } 需要根据输入满足的条件来求值。IB 题目常包括绘制此类分段函数的图像。

5. Types of Functions and Their Properties | 函数的类型及其性质

Common function types in IB Mathematics include linear (f(x) = mx + c), quadratic (f(x) = ax² + bx + c), cubic, reciprocal (f(x) = 1/x), rational, exponential (f(x) = aˣ), logarithmic (f(x) = logₐ x), and trigonometric functions. Each has distinct shapes and properties like intercepts, asymptotes, and symmetry.

IB 数学中常见的函数类型包括线性函数 (f(x) = mx + c)、二次函数 (f(x) = ax² + bx + c)、三次函数、倒数函数 (f(x) = 1/x)、有理函数、指数函数 (f(x) = aˣ)、对数函数 (f(x) = logₐ x) 以及三角函数。每种函数都有独特的形状和性质，如截距、渐近线和对称性。

An odd function satisfies f(-x) = -f(x) for all x in the domain, showing 180° rotational symmetry about the origin. An even function satisfies f(-x) = f(x) and is symmetric about the y-axis. Identifying parity helps with graphing and integrating functions.

奇函数满足对于定义域内所有 x 有 f(-x) = -f(x)，表现出关于原点的 180° 旋转对称。偶函数满足 f(-x) = f(x)，关于 y 轴对称。识别奇偶性有助于函数图像的绘制和积分运算。

6. Composite Functions | 复合函数

A composite function, written as (g ∘ f)(x) = g(f(x)), applies one function to the result of another. The order matters: f ∘ g is generally different from g ∘ f. The domain of the composite consists of all x in the domain of f such that f(x) is in the domain of g.

复合函数写作 (g ∘ f)(x) = g(f(x))，它将一个函数作用于另一个函数的结果。顺序很重要：f ∘ g 通常与 g ∘ f 不同。复合函数的定义域由所有满足 f(x) 在 g 的定义域内的 x（在 f 的定义域中）组成。

For example, if f(x) = x + 2 and g(x) = x², then (g ∘ f)(x) = (x + 2)², while (f ∘ g)(x) = x² + 2. IB examination questions frequently require computing symbolic composites or finding the domain and range of composite expressions.

例如，若 f(x) = x + 2 且 g(x) = x²，则 (g ∘ f)(x) = (x + 2)²，而 (f ∘ g)(x) = x² + 2。IB 试题经常要求计算符号化的复合函数，或求复合表达式下的定义域和值域。

7. Inverse Functions | 反函数

The inverse function f⁻¹ undoes the action of f, so that f⁻¹(f(x)) = x for all x in the domain of f, and f(f⁻¹(x)) = x for all x in the domain of f⁻¹. A function must be one-to-one (injective) over its domain to have an inverse; otherwise, we restrict the domain to make it invertible.

反函数 f⁻¹ 撤销 f 的作用，使得对于定义域内的所有 x 有 f⁻¹(f(x)) = x，且对于 f⁻¹ 定义域内的所有 x 有 f(f⁻¹(x)) = x。函数在其定义域上必须是一对一的（单射）才能有反函数；否则，我们需要限制定义域使其可逆。

To find an inverse algebraically, we swap x and y in the equation y = f(x) and then solve for y. Graphically, the inverse is a reflection of the original function across the line y = x. For instance, f(x) = 2x + 3 gives f⁻¹(x) = (x – 3)/2.

要从代数上求反函数，我们在方程 y = f(x) 中交换 x 和 y，然后解出 y。从图像上看，反函数是原函数关于直线 y = x 的反射。例如，f(x) = 2x + 3 给出 f⁻¹(x) = (x – 3)/2。

8. Graphs of Functions and Relations | 函数与关系的图像

The graph of a function is the set of points (x, f(x)) on the Cartesian plane. While functions satisfy the vertical line test, general relations may not. Graphs reveal key features: x-intercepts (zeros), y-intercepts, maximum and minimum points, asymptotes, and intervals of increase or decrease.

函数的图像是笛卡尔平面上所有点 (x, f(x)) 的集合。虽然函数满足垂直线检验，但一般关系不一定如此。图像揭示了关键特征：x 截距（零点）、y 截距、最大值和最小值点、渐近线以及增减区间。

Relations can be represented by equations like x² + y² = r² (a circle) or y² = x (a sideways parabola). These are not functions globally, but they can be split into branches, each of which is a function. Understanding the distinction is crucial for analyzing curves in IB.

关系可以用诸如 x² + y² = r²（圆）或 y² = x（侧向抛物线）的方程来表示。它们整体上不是函数，但可以拆分为分支，每个分支都是一个函数。理解这一区别对于 IB 中的曲线分析至关重要。

9. Transformations of Functions | 函数的变换

Transformations allow us to shift, stretch, compress, and reflect graphs. A function of the form y = a · f(b(x – h)) + k describes a transformation of f(x). The parameters cause: vertical stretch by factor a (and reflection over x-axis if a < 0), horizontal stretch by factor 1/|b| (and reflection over y-axis if b < 0), horizontal shift by h units, and vertical shift by k units.

变换使我们能够平移、拉伸、压缩和反射函数的图像。形如 y = a · f(b(x – h)) + k 的函数描述了 f(x) 的变换。参数分别造成：以因子 a 进行垂直拉伸（若 a < 0 则同时关于 x 轴反射），以因子 1/|b| 进行水平拉伸（若 b < 0 则关于 y 轴反射），水平平移 h 个单位，以及垂直平移 k 个单位。

In IB, you must apply transformations in the correct order: first horizontal shifts and stretches (acting on the x inside the function), then vertical stretches and shifts. For example, to graph y = 2 sin(3x – π) + 1, begin with sin x, compress horizontally by 1/3, shift right by π/3, stretch vertically by 2, and shift up by 1.

在 IB 中，你必须按照正确的顺序应用变换：先进行水平平移和拉伸（作用于函数内部的 x），然后进行垂直拉伸和平移。例如，要绘制 y = 2 sin(3x – π) + 1 的图像，从 sin x 开始，水平压缩 1/3，向右平移 π/3，垂直拉伸 2 倍，再向上平移 1。

10. Inverse Functions and Graphs | 反函数及其图像

Graphically, the inverse function is obtained by reflecting the original function across the line y = x. If a point (a, b) lies on f, then (b, a) lies on f⁻¹. This visual link helps to determine the domain and range of the inverse: the domain of f⁻¹ equals the range of f, and the range of f⁻¹ equals the domain of f.

从图像上看，反函数是通过将原函数关于直线 y = x 反射得到的。如果点 (a, b) 在 f 上，那么 (b, a) 就在 f⁻¹ 上。这种视觉联系有助于确定反函数的定义域和值域：f⁻¹ 的定义域等于 f 的值域，而 f⁻¹ 的值域等于 f 的定义域。

Not all functions have inverses over their entire domain. A quadratic function f(x) = x² is not invertible on ℝ, but if we restrict the domain to x ≥ 0, then f⁻¹(x) = √x exists. IB exercises often ask for the restricted domain that makes a function one-to-one and to subsequently find its inverse.

并非所有函数在整个定义域上都有反函数。二次函数 f(x) = x² 在 ℝ 上不可逆，但如果我们将定义域限制为 x ≥ 0，那么 f⁻¹(x) = √x 就存在了。IB 习题常要求找出使函数成为一一对应的限制定义域，并随后求出其反函数。

11. Relations Defined Implicitly and Parametrically | 隐式与参数式定义的关系

Relations need not be expressed explicitly as y = f(x). An implicit equation such as xy + sin(x + y) = 0 defines a relation that may contain multiple function branches. Parametric equations define both x and y in terms of a third variable t, e.g., x = t², y = 2t, and eliminating t often yields a Cartesian relation.

关系不一定以显式 y = f(x) 来表达。诸如 xy + sin(x + y) = 0 这样的隐式方程定义了一个可能包含多个函数分支的关系。参数方程通过第三个变量 t 来定义 x 和 y，例如 x = t², y = 2t，消去 t 通常能得到一个笛卡尔关系式。

In IB Mathematics, parametric relations are explored in the context of vector equations and kinematics. Understanding how to convert between parametric and Cartesian forms, and how to analyze the behaviour of such relations, is a key skill.

在 IB 数学中，参数关系是在向量方程和运动学的背景下探讨的。理解如何在参数形式与笛卡尔形式之间进行转换，以及如何分析这类关系的行为，是一项关键技能。

12. Applications and Problem Solving | 应用与问题求解

Relations and functions model real-world scenarios such as profit optimization, projectile motion, population growth, and radioactive decay. IB problems often combine multiple concepts: finding the maximum of a quadratic function representing profit, determining the intersection of cost and revenue functions, or solving exponential growth with a composite function.

关系和函数可以用来模拟现实世界的情景，例如利润优化、抛体运动、人口增长和放射性衰变。IB 题目通常结合多个概念：寻找表示利润的二次函数的最大值，确定成本函数与收益函数的交点，或使用复合函数求解指数增长问题。

In data-based questions, students may need to fit a function type to a scatterplot, determine its domain and range in context, and interpret parameters. Mastery of function operations enables modelling complex systems by combining simpler building blocks.

在基于数据的问题中，学生可能需要为散点图匹配一种函数类型，确定其上下文中的定义域和值域，并解释参数的含义。掌握函数运算能够通过组合更简单的构建模块来对复杂系统进行建模。

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Relations and Functions | 关系与函数