Relationships between the presidency and the Supreme Court | 总统与最高法院之间的关系：数学关系与函数

📚 Relationships between the presidency and the Supreme Court | 总统与最高法院之间的关系：数学关系与函数

The title may sound like a civics topic, but in Edexcel A-Level Mathematics we can use the relationship between the presidency and the Supreme Court as a concrete model for binary relations and functions. A relation links elements of one set to elements of another set; here we link presidents to the justices they appointed. This article will use that idea to build the key definitions and skills required for the Pure Mathematics topics of relations, functions, domain, range, inverse functions and composite functions.

这个标题听起来可能像政治学主题，但在 Edexcel A-Level 数学中，我们可以将总统与最高法院之间的关系作为二元关系与函数的具体模型。关系将一个集合中的元素与另一个集合中的元素联系起来；这里我们把总统与他们任命的大法官联系起来。本文将利用这一思路，建立纯数学中“关系、函数、定义域、值域、反函数与复合函数”等主题所需的核心定义与解题技巧。

1. Relations and Functions: Core Definitions | 关系与函数：核心定义

In Edexcel A-Level Mathematics, a relation is simply any set of ordered pairs (x, y) that connects elements of a first set X to elements of a second set Y. We write x is related to y if (x, y) belongs to the relation R. In our model, let P be the set of U.S. presidents who appointed at least one Supreme Court justice, and let J be the set of Supreme Court justices. Define R from P to J by (p, j) ∈ R if president p appointed justice j.

在 Edexcel A-Level 数学中，关系就是任意一组有序对 (x, y)，它将第一个集合 X 中的元素与第二个集合 Y 中的元素连接起来。如果 (x, y) 属于关系 R，我们就说 x 与 y 有关系。在我们的模型中，设 P 为至少任命过一名最高法院大法官的美国总统集合，设 J 为最高法院大法官集合。定义从 P 到 J 的关系 R：如果总统 p 任命了大法官 j，则 (p, j) ∈ R。

For example, if President Washington appointed Justice John Jay, then the ordered pair (Washington, Jay) is an element of R. A relation does not require every element of P to appear, and it can repeat elements of J. This flexibility is what makes relations more general than functions.

例如，如果华盛顿总统任命了约翰·杰伊大法官，那么有序对 (Washington, Jay) 就是 R 的一个元素。关系不要求 P 中的每个元素都出现，也可以重复 J 中的元素。这种灵活性使得关系比函数更一般。

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

For any relation R from X to Y, the domain is the set of all first coordinates that actually appear in R. The codomain is the whole set Y that contains possible second coordinates. The range is the set of all second coordinates that actually appear in R. In our appointment model, the domain is the set of presidents who appointed a justice, the codomain is the set of all Supreme Court justices, and the range is the set of justices who were appointed by a president in the domain.

对于从 X 到 Y 的任意关系 R，定义域是实际上出现在 R 中的所有第一坐标组成的集合。陪域是包含所有可能第二坐标的整个集合 Y。值域是实际上出现在 R 中的所有第二坐标组成的集合。在我们的任命模型中，定义域是任命过大法官的总统集合，陪域是所有最高法院大法官的集合，值域是由定义域中的总统任命的大法官集合。

In function notation, if we have f(x) = 2x + 3 with x ∈ ℝ, then the domain is all real numbers ℝ, the codomain is also ℝ unless stated otherwise, and the range is all real numbers ℝ because any real number can be obtained as 2x + 3. Being precise about domain and range is essential in Edexcel exam questions.

在函数记号中，如果 f(x) = 2x + 3 且 x ∈ ℝ，那么定义域是所有实数 ℝ，陪域如果不特别说明也是 ℝ，值域是所有实数 ℝ，因为任何实数都可以写成 2x + 3 的形式。在 Edexcel 考试题中，精确定义定义域和值域至关重要。

3. Mapping Diagrams and Arrow Representation | 映射图与箭头表示

A mapping diagram uses two ovals, one for the domain and one for the codomain, with arrows showing which elements are related. In the relation R from presidents to justices, one president can point to several justices, so the mapping is one-to-many. This is allowed for a relation, but it is not allowed for a function.

映射图使用两个椭圆形，一个表示定义域，一个表示陪域，用箭头表示哪些元素之间存在关系。在从总统到大法官的关系 R 中，一个总统可以指向多个大法官，因此这个映射是一对多的。这对关系来说是允许的，但对函数来说则不允许。

If we reverse the direction and define a relation S from J to P by (j, p) ∈ S if justice j was appointed by president p, then each justice is appointed by exactly one president, so the mapping is many-to-one. This is important because many-to-one mappings can represent functions.

如果我们反转方向，定义从 J 到 P 的关系 S：如果大法官 j 由总统 p 任命，则 (j, p) ∈ S，那么每位大法官恰好由一位总统任命，所以这个映射是多对一的。这一点很重要，因为多对一映射可以表示函数。

One-to-many: a relation, not a function.
Many-to-one: can be a function.
One-to-one: a special type of function.
Many-to-many: a relation, not a function.

一对多：是关系，不是函数。
多对一：可以是函数。
一对一：一种特殊类型的函数。
多对多：是关系，不是函数。

4. Functions as Special Relations | 函数作为特殊关系

A function f from X to Y is a relation in which every element x in the domain X is related to exactly one element y in Y. We write y = f(x). The appointment relation R from presidents to justices fails this test because a president can appoint more than one justice, so one domain element maps to many codomain elements.

从 X 到 Y 的函数 f 是一种特殊关系：定义域 X 中的每个元素 x 恰好与 Y 中的一个元素 y 相关。我们写作 y = f(x)。从总统到大法官的任命关系 R 不满足这一条件，因为一位总统可以任命多位大法官，所以一个定义域元素映射到多个陪域元素。

However, the reversed relation S from justices to presidents is a function: every justice is appointed by exactly one president. Here the domain is J, the codomain is P, and S(j) is the president who appointed justice j. This illustrates that changing the direction of a relation can turn a non-function into a function.

然而，反转后的关系 S 从大法官到总统是一个函数：每位大法官恰好由一位总统任命。这里定义域是 J，陪域是 P，S(j) 是任命大法官 j 的总统。这说明改变关系的方向可以把非函数变成函数。

5. One-to-One and Many-to-One Functions | 一一函数与多对一函数

A function f is one-to-one if different inputs always give different outputs. In symbols, if x₁ ≠ x₂ then f(x₁) ≠ f(x₂). The function f(x) = 3x + 5 is one-to-one because each output corresponds to exactly one input. A function is many-to-one if at least two different inputs give the same output. The function f(x) = x² with domain all real numbers is many-to-one because f(2) = 4 and f(−2) = 4.

如果不同的输入总是得到不同的输出，那么函数 f 就是一一函数。用符号表示：如果 x₁ ≠ x₂，那么 f(x₁) ≠ f(x₂)。函数 f(x) = 3x + 5 是一一函数，因为每个输出恰好对应一个输入。如果至少两个不同的输入得到相同的输出，那么函数就是多对一函数。函数 f(x) = x² 以所有实数为定义域时是多对一的，因为 f(2) = 4 且 f(−2) = 4。

Knowing whether a function is one-to-one is essential for inverse functions. In Edexcel Pure Mathematics, you must be able to identify one-to-one and many-to-one functions from their graphs, equations and mapping diagrams.

判断一个函数是否是一一函数对反函数至关重要。在 Edexcel 纯数学中，你必须能够从图像、方程和映射图识别一一函数和多对一函数。

6. Composite Functions | 复合函数

A composite function is formed when one function is applied to the result of another. If f and g are functions, the composite function fg is defined by fg(x) = f(g(x)). The output of g becomes the input of f. The order matters: fg(x) is generally different from gf(x) unless the functions commute.

当一个函数作用于另一个函数的结果时，就形成了复合函数。如果 f 和 g 是函数，复合函数 fg 定义为 fg(x) = f(g(x))。g 的输出成为 f 的输入。顺序很重要：除非函数可以交换，否则 fg(x) 通常与 gf(x) 不同。

For example, if f(x) = 2x and g(x) = x + 3, then fg(x) = f(g(x)) = f(x + 3) = 2(x + 3) = 2x + 6. The reverse gf(x) = g(f(x)) = g(2x) = 2x + 3, which is different. In our analogy, we might apply one relation to find a president from a justice, then another relation to find the justice’s confirmation year, creating a chain of related elements.

例如，如果 f(x) = 2x，g(x) = x + 3，那么 fg(x) = f(g(x)) = f(x + 3) = 2(x + 3) = 2x + 6。反向 gf(x) = g(f(x)) = g(2x) = 2x + 3，两者不同。在我们的类比中，可以先从一个关系找到任命某位大法官的总统，再应用另一个关系找到该大法官的确认年份，形成一条相关元素的链条。

7. Inverse Functions | 反函数

A function f has an inverse function f⁻¹ if and only if f is one-to-one. The inverse function reverses the effect of f, so that f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. To find the inverse of a one-to-one function, write y = f(x), rearrange to make x the subject, then swap x and y.

当且仅当函数 f 是一一函数时，它才具有反函数 f⁻¹。反函数会逆转 f 的作用，因此 f⁻¹(f(x)) = x 且 f(f⁻¹(x)) = x。求一一函数的反函数时，先写出 y = f(x)，将 x 变成主项，然后交换 x 和 y。

For example, if f(x) = 3x − 7, then y = 3x − 7. Rearranging gives x = (y + 7) ÷ 3. Swapping x and y gives f⁻¹(x) = (x + 7) ÷ 3. The domain of f becomes the range of f⁻¹, and the range of f becomes the domain of f⁻¹.

例如，如果 f(x) = 3x − 7，那么 y = 3x − 7。整理得 x = (y + 7) ÷ 3。交换 x 和 y 得 f⁻¹(x) = (x + 7) ÷ 3。f 的定义域成为 f⁻¹ 的值域，f 的值域成为 f⁻¹ 的定义域。

f(f⁻¹(x)) = x and f⁻¹(f(x)) = x

8. Graphs and the Vertical Line Test | 图像与垂线检验

For a graph to represent a function, every vertical line drawn through the domain must cut the graph at most once. This is called the vertical line test. If a vertical line cuts a graph more than once, the graph represents a relation but not a function.

如果一条曲线表示函数，那么穿过定义域的任意垂线最多只能与图像相交一次。这称为垂线检验。如果一条垂线与图像相交超过一次，那么该图像表示关系而不是函数。

A horizontal line test is used to check whether a function is one-to-one. If every horizontal line cuts the graph at most once, the function is one-to-one and therefore has an inverse. The graph of y = x² fails the horizontal line test unless the domain is restricted to x ≥ 0.

水平线检验用于判断函数是否是一一函数。如果每条水平线最多与图像相交一次，那么这个函数就是一一函数，因此存在反函数。y = x² 的图像无法通过水平线检验，除非将定义域限制为 x ≥ 0。

Vertical line test: checks if a graph is a function.
Horizontal line test: checks if a function is one-to-one.

垂线检验：判断图像是否为函数。
水平线检验：判断函数是否为一对一。

9. Worked Example: Relations to Functions | 例题：从关系到函数

A relation R is defined by R = {(1, 2), (2, 4), (3, 6), (4, 8)}. State the domain and range of R. Determine whether R is a function. If it is a function, state whether it is one-to-one and find its inverse.

一个关系 R 定义为 R = {(1, 2), (2,

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Relationships between the presidency and the Supreme Court | 总统与最高法院之间的关系：数学关系与函数