Mastering Functions: Concepts and Fundamental Properties | 函数概念与基本性质全面掌握

📚 Mastering Functions: Concepts and Fundamental Properties | 函数概念与基本性质全面掌握

Functions are one of the most essential topics in mathematics, forming the foundation for algebra, calculus, and applied sciences. Understanding their definitions, domains, and properties is critical for solving exam problems accurately.

函数是数学中最核心的主题之一，是所有代数、微积分及应用科学的基石。准确理解函数的定义、定义域及性质，是在考试中顺利解题的关键。

1. Definition of a Function | 函数的定义

A function is a relation between two sets, usually called the domain and codomain, such that each input in the domain is assigned to exactly one output in the codomain. This “one-to-one or many-to-one” rule distinguishes a function from a general relation.

函数是两个集合之间的一种对应关系，通常称为定义域和陪域。定义域中的每一个输入都唯一对应陪域中的一个输出。这种“一对一或多对一”的规则，使函数区别于一般的“关系”。

For example, the equation y = 2x + 1 defines a function because for every real number x, there is exactly one y. However, the equation x² + y² = 1 does not define a function over all real x, because a single x can produce two different y values.

例如，方程 y = 2x + 1 定义了一个函数，因为对每个实数 x，有且仅有一个 y。但方程 x² + y² = 1 在全体实数范围内不是函数，因为同一个 x 可能对应两个不同的 y 值。

In exams, the vertical line test is often used: a graph represents a function if and only if every vertical line intersects the graph at no more than one point.

考试中常用“竖直线检验”：一个图像表示函数，当且仅当每条竖直线与图像至多交于一点。

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 set of all possible output values produced by the function.

定义域是函数所有可能输入值的集合；值域是函数所有可能输出值的集合。

To find the domain, look for restrictions: denominators cannot be zero, radicands of even roots must be non-negative, and arguments of logarithms must be positive.

求定义域时需注意限制条件：分母不能为零，偶次根号下的式子必须非负，对数的真数必须为正。

Exampleː For f(x) = 1/(x − 3), the domain is all real numbers except x = 3. Its range is all real numbers except y = 0.

例：对于 f(x) = 1/(x − 3)，定义域为除去 x = 3 以外的所有实数，值域为除去 y = 0 以外的所有实数。

When finding the range, it is often helpful to solve for x in terms of y and then determine the allowed y values.

求值域时，常见方法是将 x 用 y 表示，再确定 y 可取的取值范围。

3. Representations of Functions | 函数的表示法

Functions can be represented in four main ways: verbal (words), numerical (tables), algebraic (formulas), and graphical (graphs). Each representation highlights different aspects of the function.

函数主要有四种表示方式：文字描述、数值表格、代数公式和图像。每种表示法从不同侧面反映函数的特征。

For example, a table of values can reveal patterns, while an algebraic formula allows precise calculation. A graph provides a visual understanding of monotonicity, symmetry, and asymptotes.

例如，数值表能揭示规律，代数公式便于精确计算，图像则直观展示单调性、对称性和渐近线等特征。

Verbal: “The cost is 5 dollars per item plus a fixed fee of 10 dollars.”
Algebraic: C(n) = 5n + 10
Table: n=0,1,2 → C=10,15,20
Graph: a straight line with slope 5 and y-intercept 10

In any examination, you must be able to switch between these representations efficiently.

在考试中，你需要能够熟练地在各种表示法之间转换。

4. Monotonicity | 单调性

A function is increasing on an interval if, for any x₁ < x₂ in that interval, f(x₁) ≤ f(x₂). It is strictly increasing if f(x₁) < f(x₂). Decreasing and strictly decreasing are defined analogously.

若在某个区间内，对任意 x₁ < x₂，都有 f(x₁) ≤ f(x₂)，则称函数在该区间上递增；若总有 f(x₁) < f(x₂)，则称为严格递增。递减与严格递减的定义类似。

To determine monotonicity, use the derivative: if f'(x) > 0 on an interval, the function is increasing there; if f'(x) < 0, it is decreasing. For discrete or piecewise functions, compare consecutive values.

判断单调性常用导数：若在某区间内 f'(x) > 0，则函数递增；若 f'(x) < 0，则递减。对于分段的离散函数，可比较相邻取值。

Monotonicity is a key property for finding inverse functions and determining the existence of roots.

单调性是求反函数以及判断方程实根存在性的重要依据。

5. Even and Odd Functions | 奇偶性

A function f(x) is even if f(−x) = f(x) for all x in its domain. Its graph is symmetric about the y-axis. Examples include x² and cos x.

若对于定义域内所有 x，都有 f(−x) = f(x)，则称 f(x) 为偶函数。偶函数的图像关于 y 轴对称，例如 x² 和 cos x。

A function is odd if f(−x) = −f(x) for all x in its domain. Its graph is symmetric about the origin. Examples include x³ and sin x.

若对于定义域内所有 x，都有 f(−x) = −f(x)，则称 f(x) 为奇函数。奇函数的图像关于原点对称，例如 x³ 和 sin x。

Many functions are neither even nor odd, such as 2x + 1. However, every function can be written as the sum of an even function and an odd function.

许多函数既不是偶函数也不是奇函数，例如 2x + 1。不过，任意函数都可以表示为一个偶函数与一个奇函数之和。

When solving exams, first check whether the domain itself is symmetric about zero; otherwise the function cannot be even or odd.

解题时，先要看定义域是否关于原点对称；若不对称，则函数不可能具有奇偶性。

6. Periodicity | 周期性

A function f(x) is periodic with period T > 0 if f(x + T) = f(x) for all x in its domain. The smallest positive value of T is called the fundamental period.

若存在正数 T，使得对定义域内所有 x，都有 f(x + T) = f(x)，则称 f(x) 为周期函数。使等式成立的最小正数 T 称为最小正周期。

Trigonometric functions are the most common examples: sin x and cos x have period 2π, while tan x has period π.

三角函数是最常见的周期函数：sin x 和 cos x 的周期为 2π，tan x 的周期为 π。

If two periodic functions have different periods, the period of their sum is the least common multiple of the two periods, provided such a multiple exists.

若两个周期函数具有不同周期，它们之和的周期为这两个周期的最小公倍数（如果该公倍数存在）。

Be careful: constant functions are periodic with any period, but they do not possess a smallest positive period.

注意：常数函数以任意正数为周期，但它没有最小正周期。

7. Boundedness | 有界性

A function is bounded above on an interval if there exists a number M such that f(x) ≤ M for all x in that interval. It is bounded below if there exists m such that f(x) ≥ m. A function is bounded if it is bounded both above and below.

如果存在常数 M，使得某区间内所有 x 都满足 f(x) ≤ M，则称函数在该区间上有上界；若存在 m 使 f(x) ≥ m，则称有下界。既有上界又有下界的函数称为有界函数。

For example, sin x and cos x are bounded because −1 ≤ sin x ≤ 1 and −1 ≤ cos x ≤ 1. The function 1/x is unbounded on (0, 1) but bounded on [1, 2].

例如，sin x 和 cos x 是有界的，因为 −1 ≤ sin x ≤ 1，−1 ≤ cos x ≤ 1。函数 1/x 在区间 (0, 1) 上无界，但在 [1, 2] 上有界。

In exam problems, boundedness is often used to determine the range or to prove the existence of maxima and minima.

在考试题目中，有界性常用于确定值域或证明最大值、最小值的存在性。

8. Inverse Functions | 反函数

If a function f is one-to-one (injective), it has an inverse function f⁻¹, defined by f⁻¹(y) = x if and only if f(x) = y. The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.

若函数 f 是一一对应的（单射），则它存在反函数 f⁻¹，其定义为：当且仅当 f(x) = y 时，f⁻¹(y) = x。f⁻¹ 的定义域是 f 的值域，f⁻¹ 的值域是 f 的定义域。

Graphically, the graph of f⁻¹ is the reflection of the graph of f across the line y = x. Algebraically, to find f⁻¹, we solve y = f(x) for x and then swap x and y.

图像上，f⁻¹ 的图像是 f 的图像关于直线 y = x 的对称图形。代数上，求反函数时先由 y = f(x) 解出 x，再交换 x 与 y。

A function must be strictly monotonic on its domain in order to have an inverse that is also a function.

一个函数必须在其定义域上严格单调，才能保证反函数也是函数。

Exampleː For f(x) = 2x + 1, solving y = 2x + 1 gives x = (y − 1)/2, so f⁻¹(x) = (x − 1)/2.

例：对 f(x) = 2x + 1，由 y = 2x + 1 解得 x = (y − 1)/2，因此 f⁻¹(x) = (x − 1)/2。

9. Composite Functions | 复合函数

The composite of two functions f and g, written as (f ∘ g)(x) = f(g(x)), means that we first apply g to x, then apply f to the result. The domain of the composite consists of all x in the domain of g such that g(x) lies in the domain of f.

函数 f 和 g 的复合记作 (f ∘ g)(x) = f(g(x))，即先对 x 施加 g，再对结果施加 f。复合函数的定义域是 g 定义域中能使 g(x) 落在 f 定义域内的那些 x 的全体。

For example, if f(x) = √x and g(x) = x − 1, then (f ∘ g)(x) = √(x − 1). Its domain is x ≥ 1, because √(x − 1) requires x − 1 ≥ 0.

例如，若 f(x) = √x，g(x) = x − 1，则 (f ∘ g)(x) = √(x − 1)。其定义域为 x ≥ 1，因为根号内需要 x − 1 ≥ 0。

Remember that in general f ∘ g ≠ g ∘ f. When taking exams, always pay attention to the order of operations and the resulting domain restrictions.

请记住，一般 f ∘ g ≠ g ∘ f。考试时务必注意运算顺序以及由此产生的定义域限制。

10. Basic Elementary Functions and Transformations | 基本初等函数与图像变换

The basic elementary functions include constant functions, power functions, exponential functions, logarithmic functions, trigonometric functions, and inverse trigonometric functions.

基本初等函数包括常函数、幂函数、指数函数、对数函数、三角函数和反三角函数。

Graph transformations are a frequent exam topic. Vertical shifts: y = f(x) + k moves the graph up or down. Horizontal shifts: y = f(x − h) moves the graph right or left. Reflections: y = −f(x) reflects over the x-axis; y = f(−x) reflects over the y-axis.

图像变换是高频考点。纵向平移：y = f(x) + k 使图像上下移动；横向平移：y = f(x − h) 使图像左右移动；反射：y = −f(x) 关于 x 轴对称，y = f(−x) 关于 y 轴对称。

Stretching and compression: y = a f(x) stretches vertically if |a| > 1 and compresses if 0 < |a| < 1. Similarly, y = f(bx) compresses horizontally if |b| > 1 and stretches if 0 < |b| < 1.

伸缩变换：若 |a| > 1，则 y = a f(x) 纵向拉伸；若 0 < |a| < 1，则纵向压缩。类似地，若 |b| > 1，则 y = f(bx) 横向压缩；若 0 < |b| < 1，则横向拉伸。

A useful order for applying transformations is: horizontal shift, reflection/stretch (on x), then vertical stretch/reflection, and finally vertical shift.

一个实用的变换顺序是：先横向平移，再进行关于 x 的伸缩/反射，然后进行纵向伸缩/反射，最后纵向平移。

11. Common Exam Mistakes and Summary | 常见易错点与总结

Many students lose marks due to ignoring the domain before simplifying, forgetting to check one-to-one before finding an inverse, and confusing (f ∘ g)(x) with f(x) · g(x).

很多学生失分是因为在化简前忽略了定义域、求反函数前未检验一一对应、以及混淆复合函数 f(g(x)) 与乘积 f(x)·g(x)。

Let us summarize the key points:

让我们总结关键考点：

Always determine the natural domain before analyzing any function.
Check monotonicity using derivatives or interval comparisons.
Verify symmetry conditions before classifying even/odd.
Use both algebraic and graphical perspectives for inverse functions.
Apply transformations in the correct order.

分析任何函数前，先求出其自然定义域。
用导数或区间比较判断单调性。
判断奇偶性前，先验证对称条件。
对反函数要同时从代数和图像角度理解。
按正确顺序应用图像变换。

Mastering these fundamental concepts will strongly support your performance in algebra and calculus examinations.

掌握这些基本概念，将极大提升你在代数与微积分考试中的表现。

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Mastering Functions: Concepts and Fundamental Properties | 函数概念与基本性质全面掌握