Functions: Concepts and Properties Explained | 函数的概念与性质解析

📚 Functions: Concepts and Properties Explained | 函数的概念与性质解析

A function is one of the most fundamental concepts in mathematics. It describes a special relationship between two sets: every input corresponds to exactly one output. Understanding functions is essential for success in algebra, calculus, and beyond.

函数是数学中最基本的概念之一。它描述了两个集合之间的一种特殊关系：每一个输入都恰好对应一个输出。理解函数是学好代数、微积分乃至更高级数学的关键基础。

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1. Definition of a Function | 函数的定义

A function f from set A to set B, written as f: A → B, assigns each element x in set A to exactly one element y in set B. The set A is called the domain, and the set of all outputs is called the range.

函数是从集合A到集合B的一种对应关系，记作 f: A → B，它将集合A中的每一个元素x都唯一地对应到集合B中的一个元素y。集合A称为定义域，所有输出值构成的集合称为值域。

f: A → B, x ↦ y = f(x)

For example, the function f(x) = x² maps 2 to 4, −3 to 9, and so on. Each input has only one output, which is the defining property of a function.

例如，函数 f(x) = x² 将2映射到4，将−3映射到9，依此类推。每个输入只有一个输出，这是函数的定义性特征。

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2. Domain and Range | 定义域与值域

The domain of a function is the complete set of possible input values. The range is the complete set of possible output values produced by the function. Determining these sets is often the first step in analysing any function.

函数的定义域是所有可能输入值的完整集合。值域是函数产生的所有可能输出值的完整集合。确定这两个集合通常是分析任何函数的第一步。

Function | 函数	Domain | 定义域	Range | 值域
f(x) = √x	x ≥ 0	y ≥ 0
f(x) = 1/x	x ≠ 0	y ≠ 0
f(x) = x²	All real numbers	y ≥ 0

When finding the domain, look for restrictions such as division by zero or square roots of negative numbers. The range can often be found by considering the minimum or maximum value of the function.

求定义域时，需要注意各种限制条件，如分母不能为零，偶次根号下不能为负数。值域通常可以通过考虑函数的最小值或最大值来确定。

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

The notation f(x) is read as “f of x” and represents the output of the function f when the input is x. This notation is powerful because it allows us to substitute values, expressions, or even other functions.

记号 f(x) 读作”f of x”，表示当输入为x时函数f的输出。这种记号非常强大，因为它允许我们代入数值、表达式甚至是其他函数。

• If f(x) = 3x + 2, then f(4) = 3(4) + 2 = 14.

  如果 f(x) = 3x + 2，那么 f(4) = 3(4) + 2 = 14。

• If f(x) = x² − 1, then f(a + 1) = (a + 1)² − 1 = a² + 2a.

  如果 f(x) = x² − 1，那么 f(a + 1) = (a + 1)² − 1 = a² + 2a。

Function notation also helps us understand composite functions, where the output of one function becomes the input of another.

函数记号还帮助我们理解复合函数，即一个函数的输出成为另一个函数的输入。

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4. One-to-One and Onto Functions | 单射与满射函数

A one-to-one (injective) function maps distinct inputs to distinct outputs. In other words, if f(a) = f(b), then a must equal b. An onto (surjective) function covers every element of the codomain.

单射函数（一一对应）将不同的输入映射到不同的输出。换句话说，如果 f(a) = f(b)，则必有 a = b。满射函数（映上）覆盖了陪域中的每一个元素。

f(x₁) = f(x₂) ⇒ x₁ = x₂ (one-to-one)

The horizontal line test is a quick way to check if a function is one-to-one: if any horizontal line intersects the graph more than once, the function is not injective.

水平线检验法是判断单射的快捷方法：如果任何水平直线与图像相交超过一次，则该函数不是单射。

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5. Even and Odd Functions | 偶函数与奇函数

An even function satisfies f(−x) = f(x) for all x in its domain. Its graph is symmetric about the y-axis. An odd function satisfies f(−x) = −f(x), and its graph is symmetric about the origin.

偶函数满足 f(−x) = f(x) 对于定义域内的所有x成立。其图像关于y轴对称。奇函数满足 f(−x) = −f(x)，其图像关于原点对称。

Even | 偶函数	Odd | 奇函数
f(−x) = f(x)	f(−x) = −f(x)
Symmetric about y-axis | 关于y轴对称	Symmetric about origin | 关于原点对称
Examples: x², cos(x), |x|	Examples: x³, sin(x), tan(x)

Many functions are neither even nor odd. For example, f(x) = 2x + 1 fails both tests.

许多函数既不是偶函数也不是奇函数。例如，f(x) = 2x + 1 两个测试都不满足。

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6. Periodic Functions | 周期函数

A function is periodic with period T if f(x + T) = f(x) for all x in its domain, where T is a positive constant. The smallest positive value of T is called the fundamental period.

如果函数满足 f(x + T) = f(x) 对其定义域内所有x成立，其中T为正常数，则该函数是周期函数。使等式成立的最小正数T称为最小正周期。

f(x + T) = f(x), T > 0

Trigonometric functions are the most common periodic functions. For example, sin(x) and cos(x) have a fundamental period of 2π, while tan(x) has a period of π.

三角函数是最常见的周期函数。例如，sin(x) 和 cos(x) 的最小正周期为2π，而 tan(x) 的周期为π。

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7. Increasing and Decreasing Functions | 增函数与减函数

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. These properties describe the monotonic behaviour of a function.

如果对于区间内任意 x₂ > x₁，都有 f(x₂) > f(x₁)，则函数在该区间上是增函数。如果有 f(x₂) < f(x₁)，则是减函数。这些性质描述了函数的单调性。

• f(x) = 2x is increasing on its entire domain.

  f(x) = 2x 在其整个定义域上是增函数。

• f(x) = −x³ is decreasing on its entire domain.

  f(x) = −x³ 在其整个定义域上是减函数。

• f(x) = x² is decreasing on (−∞, 0) and increasing on (0, ∞).

  f(x) = x² 在 (−∞, 0) 上是减函数，在 (0, ∞) 上是增函数。

Derivatives provide a powerful tool for determining where a function is increasing or decreasing: if f′(x) > 0, the function is increasing; if f′(x) < 0, it is decreasing.

导数是判断函数增减性的强大工具：若 f′(x) > 0，函数递增；若 f′(x) < 0，函数递减。

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8. Bounded Functions | 有界函数

A function f is bounded above if there exists a real number M such that f(x) ≤ M for all x in its domain. It is bounded below if there exists m such that f(x) ≥ m. A function bounded both above and below is simply called bounded.

如果存在实数M使得对于定义域内所有x都有 f(x) ≤ M，则函数有上界。如果存在m使得 f(x) ≥ m，则函数有下界。上下界都存在的函数称为有界函数。

m ≤ f(x) ≤ M for all x ∈ Domain

For example, f(x) = sin(x) is bounded because −1 ≤ sin(x) ≤ 1. In contrast, f(x) = x² is unbounded above because it grows without limit.

例如，f(x) = sin(x) 是有界的，因为 −1 ≤ sin(x) ≤ 1。相比之下，f(x) = x² 无上界，因为它的值无限增长。

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9. Composite Functions | 复合函数

Given two functions f and g, the composite function (f ∘ g)(x) = f(g(x)) is formed by applying g first, then applying f to the result. The domain of the composite function consists of all x in the domain of g such that g(x) is in the domain of f.

给定两个函数f和g，复合函数 (f ∘ g)(x) = f(g(x)) 是先将g作用于x，再将f作用于其结果而形成的。复合函数的定义域由g定义域中所有满足 g(x) 属于f定义域的x组成。

For example, if f(x) = x² and g(x) = x + 1, then:

例如，如果 f(x) = x²，g(x) = x + 1，那么：

(f ∘ g)(x) = f(g(x)) = (x + 1)² = x² + 2x + 1

Note that composition is not commutative: f ∘ g ≠ g ∘ f in general.

请注意，复合运算不满足交换律：一般情况下 f ∘ g ≠ g ∘ f。

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10. Inverse Functions | 反函数

The inverse of a function f, denoted f⁻¹, reverses the effect of f. If f(a) = b, then f⁻¹(b) = a. A function has an inverse if and only if it is one-to-one.

函数f的反函数，记作 f⁻¹，是逆转f作用的函数。如果 f(a) = b，则 f⁻¹(b) = a。一个函数存在反函数当且仅当它是单射。

f(f⁻¹(x)) = f⁻¹(f(x)) = x

To find the inverse of a function algebraically: replace f(x) with y, swap x and y, then solve for y. The graph of f⁻¹ is the reflection of the graph of f across the line y = x.

代数上求反函数的步骤为：将 f(x) 替换为 y，交换x和y，然后解出y。函数f⁻¹的图像是f的图像关于直线 y = x 的反射。

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11. Graphical Transformations | 图像变换

Understanding how transformations affect the graph of a function is crucial. Horizontal shifts move the graph left or right, vertical shifts move it up or down, and reflections flip it across an axis.

理解变换如何影响函数图像至关重要。水平平移使图像左右移动，垂直平移使图像上下移动，反射则使图像关于某轴对称翻转。

Transformation | 变换	Effect on Graph | 对图像的影响
y = f(x) + k	Shift up by k units | 上移k个单位
y = f(x + h)	Shift left by h units | 左移h个单位
y = −f(x)	Reflect across x-axis | 关于x轴对称
y = f(−x)	Reflect across y-axis | 关于y轴对称
y = a·f(x), a > 1	Vertical stretch | 纵向拉伸

These transformations are essential for sketching graphs efficiently without plotting individual points.

这些变换对于无需逐点描画即可高效绘制函数图像至关重要。

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12. The Horizontal Line Test and Invertibility | 水平线检验与可逆性

The horizontal line test is a graphical method to determine whether a function is one-to-one and therefore has an inverse. If every horizontal line intersects the graph at most once, the function is one-to-one.

水平线检验是一种图形化方法，用于判断函数是否为单射，从而判断其是否存在反函数。如果每条水平直线与图像最多相交一次，则该函数是单射。

For instance, f(x) = x² fails the horizontal line test because the horizontal line y = 4 intersects the graph at both x = 2 and x = −2. However, if we restrict the domain to x ≥ 0, it becomes one-to-one and invertible.

例如，f(x) = x² 不通过水平线检验，因为水平线 y = 4 在 x = 2 和 x = −2 两点与图像相交。然而，如果将定义域限制为 x ≥ 0，它就变成单射并可逆了。

f(x) = x², x ≥ 0 ⇒ f⁻¹(x) = √x

This example highlights the importance of domain restrictions when working with inverse functions.

这个例子凸显了在处理反函数时限制定义域的重要性。

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Mastering the concept of functions and their properties is a cornerstone of mathematical study. Each property — whether it be domain and range, monotonicity, symmetry, or invertibility — provides a lens through which we can better understand the behaviour of mathematical relationships. With these tools, students are well-equipped to tackle more advanced topics in algebra, calculus, and beyond.

掌握函数的概念及其性质是数学学习的基石。每一个性质——无论是定义域与值域、单调性、对称性还是可逆性——都为我们提供了理解数学关系行为的视角。掌握这些工具后，学生将有能力应对代数、微积分以及更高级数学主题的挑战。

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