📚 Domain and Range | 定义域与值域
Whenever we work with functions, we are interested in what can go into the function and what comes out. The set of all possible inputs is called the domain, and the set of all possible outputs is called the range. Mastering domain and range is essential for understanding graphs, equations, and transformations in IGCSE Mathematics.
我们在处理函数时，总想知道函数能接受哪些输入，又会产生哪些输出。所有可能输入的集合称为定义域，所有可能输出的集合称为值域。掌握定义域和值域是理解图像、方程和变换的基础，也是IGCSE数学的重要考点。
1. Understanding Functions and Their Inputs/Outputs | 理解函数及其输入与输出
A function is a rule that assigns exactly one output to each input. In set terms, it is a mapping from a set of inputs (the domain) to a set of outputs (the range). Every x-value in the domain produces one and only one f(x) value.
函数是一个规则，为每个输入分配恰好一个输出。用集合的语言来说，它是一个从输入集（定义域）到输出集（值域）的映射。定义域中的每一个 x 值都产生唯一一个 f(x) 值。
represents any element of the domain, and f(x) represents the corresponding element of the range. The domain is often all real numbers unless a restriction is specified.
当我们写出 f(x) = 2x + 3 时，字母 x 代表定义域中的任意元素，f(x) 则代表值域中对应的元素。除非有特别限制，定义域通常为所有实数。
2. What is Domain? | 什么是定义域？
The domain of a function is the complete set of possible values of the independent variable (usually x) for which the function is defined. For a function to be defined, the expression must make mathematical sense – no division by zero, no square roots of negative numbers in the real number system.
函数的定义域是使函数有意义的所有自变量（通常为 x）取值的集合。要使函数有定义，表达式必须在数学上有意义——在实数系中，分母不能为零，平方根内的数不能为负。
x² + 1, with no further context, the domain is usually assumed to be all real numbers, written as {x : x ∈ ℝ}.
如果函数以 f(x) = x² + 1 给出，而没有额外的限制，那么定义域通常默认为所有实数，记为 {x : x ∈ ℝ}。
3. What is Range? | 什么是值域？
The range is the set of all possible output values (y-values or f(x)-values) that the function actually produces when the domain is substituted into it. The range depends heavily on the domain and the type of function.
值域是指将定义域中的值代入函数后，函数实际产生的所有输出值（y 值或 f(x) 值）的集合。值域高度依赖于定义域和函数的类型。
≥ 0}. However, if the domain were restricted to x ≥ 2, the range would become {y : y ≥ 4}.
对于 f(x) = x²，若定义域为所有实数，输出值不会为负，因此值域为 {y : y ≥ 0}。但如果定义域限制为 x ≥ 2，那么值域就变为 {y : y ≥ 4}。
4. Representing Domain and Range Using Set Notation | 用集合符号表示定义域与值域
In IGCSE, you are expected to use set notation such as {x : x > 5} to describe domain and range. The vertical line or colon means 'such that'. We also often specify that x belongs to the set of real numbers (ℝ).
在IGCSE中，你需要会用集合符号来描述定义域和值域，例如 {x : x > 5}。其中的竖线或冒号表示“满足……”。我们通常还会指出 x 属于实数集 ℝ。
Common notations include {x : x ∈ ℝ, x ≠ 2} for all real numbers except 2, and {y : y ∈ ℝ, y ≥ 0} for non‑negative Real numbers. Inequalities and set membership are combined for precision.
常见的写法包括 {x : x ∈ ℝ, x ≠ 2} 表示除 2 以外的所有实数，{y : y ∈ ℝ, y ≥ 0} 表示非负实数。把不等式和属于符号组合在一起，能让表达更精确。
| Example function | Domain (set notation) | Range (set notation) |
|---|---|---|
| f(x) = 2x | {x : x ∈ ℝ} | {y : y ∈ ℝ} |
| f(x) = x² | {x : x ∈ ℝ} | {y : y ∈ ℝ, y ≥ 0} |
| f(x) = 1/x | {x : x ∈ ℝ, x ≠ 0} | {y : y ∈ ℝ, y ≠ 0} |
| f(x) = √x-3 | {x : x ∈ ℝ, x ≥ 3} | {y : y ∈ ℝ, y ≥ 0} |
Sometimes you will see interval notation like (2, ∞) or [0, 1], but set‑builder form using braces and 'such that' is the primary method required at this level.
有时候你会看到区间符号，例如 (2, ∞) 或 [0, 1]，但在现阶段，主要要求的还是用花括号和“满足……”构建的集合表示法。
5. Finding Domain from Equations (Restrictions) | 从方程寻找定义域（限制条件）
To find the domain algebraically, look for any operations that restrict input values. The two most common restrictions in IGCSE are denominators and even roots (usually square roots).
要代数化地求定义域，就要找出哪些运算会限制输入值。IGCSE中最常见的两种限制是分母和偶次根式（通常是平方根）。
ominator must not be zero. So for f(x) = 1/(x − 5), we set x − 5 ≠ 0, giving domain {x : x ∈ ℝ, x ≠ 5}.
对于分母：分母绝不能为零。因此对于 f(x) = 1/(x − 5)，要令 x − 5 ≠ 0，得到定义域 {x : x ∈ ℝ, x ≠ 5}。
any even root) in the real numbers, the expression inside the root must be ≥ 0. For g(x) = √2x + 8, solve 2x + 8 ≥ 0 → x ≥ −4, so the domain is {x : x ∈ ℝ, x ≥ −4}.
对于平方根（或任何偶次根），在实数范围内根号内的表达式必须 ≥ 0。对于 g(x) = √2x + 8，解 2x + 8 ≥ 0 → x ≥ −4，所以定义域为 {x : x ∈ ℝ, x ≥ −4}。
nd a square root, you must satisfy both conditions simultaneously. For instance, h(x) = 1/√x − 2 requires x − 2 > 0 (strictly greater because of the denominator), yielding domain {x : x > 2}.
如果一个函数同时含有分母和平方根，就需要同时满足两个条件。例如 h(x) = 1/√x − 2 要求 x − 2 > 0（严格大于，因为它在分母上），得到定义域 {x : x > 2}。
6. Domain of Square Root Functions | 平方根函数的定义域
Square root functions have a built‑in restriction: the radicand (the expression under the square root) must be greater than or equal to zero. This is because the square root of a negative number is not a real number.
平方根函数天然有一个限制：被开方数（根号下的表达式）必须大于等于零。这是因为负数的平方根在实数范围内是没有定义的。
For f(x) = √x + 4 uare root function starts at the point where the radicand is zero and extends to the right.
对于 f(x) = √x + 4，需要 x + 4 ≥ 0，因此 x ≥ −4。定义域写作 {x : x ≥ −4}。平方根函数的图像从被开方数为零的点开始，向右延伸。
appears in the denominator, the radicand must be strictly greater than zero. For g(x) = 1/√x − 1, we require x − 1 > 0, so x > 1. Domain: {x : x > 1}.
当平方根出现在分母时，被开方数必须严格大于零。对于 g(x) = 1/√x − 1，要求 x − 1 > 0，即 x > 1。定义域为 {x : x > 1}。
7. Domain of Rational Functions | 有理函数的定义域
+ 2)/(x² − 9). The domain is all real numbers except those that make the denominator zero.
有理函数是两个多项式的比，例如 f(x) = (x + 2)/(x² − 9)。其定义域是所有使分母不为零的实数。
equal to zero and solve. For x² − 9 = 0, we get x = 3 or x = −3. Thus the domain is {x : x ∈ ℝ, x ≠ 3, x ≠ −3}.
要找出被排除的值，令分母等于零并求解。对于 x² − 9 = 0，得到 x = 3 或 x = −3。因此定义域为 {x : x ∈ ℝ, x ≠ 3, x ≠ −3}。
Even if a factor cancels with the numerator, the domain restriction remains because the original function is undefined at that point. Always consider the original form of the function when finding the domain.
即使某个因式与分子约掉了，定义域的限制依然存在，因为原函数在该点没有定义。求定义域时，始终要考虑函数的原有形式。
8. Determining Range from Graphs | 从图像确定值域
One of the most reliable ways to find the range is to sketch or visualise the graph of the function. The range is the set of all y‑values that the graph covers as x runs through the domain.
寻找值域最可靠的方法之一就是画出或想象函数的图像。值域就是当 x 取遍定义域时，图像覆盖的所有 y 值的集合。
For a horizontal line like y = 3, the range is simply {3}. For a linear function with non‑zero slope, the range is all real numbers {y : y ∈ ℝ}.
对于像 y = 3 这样的水平线，值域就是 {3}。对于斜率不为零的一次函数，值域是全体实数 {y : y ∈ ℝ}。
= x² − 4x + 1, find the vertex. Completing the square gives (x − 2)² − 3, so the minimum y‑value is −3. The range is {y : y ≥ −3}.
对于开口向上的抛物线，例如 y = x² − 4x + 1，先找到顶点。通过配方得到 (x − 2)² − 3，因此最小 y 值为 −3。值域为 {y : y ≥ −3}。
ain {x : −1 ≤ x ≤ 2}, the minimum y is 0 and maximum y is 4, giving range {y : 0 ≤ y ≤ 4}.
如果定义域有明确限制，只使用图像在该定义域上的部分求值域。对于 y = x²，定义域为 {x : −1 ≤ x ≤ 2} 时，最小 y 值为 0，最大为 4，值域为 {y : 0 ≤ y ≤ 4}。
9. Range of Linear and Quadratic Functions | 线性与二次函数的值域
s a straight line graph that extends infinitely in both vertical directions, so its range is all real numbers, {y : y ∈ ℝ}.
线性函数 f(x) = mx + c (m ≠ 0) 的图像是一条向两个方向无限延伸的直线，因此它的值域是全体实数 {y : y ∈ ℝ}。
x + c, the graph is a parabola. The range depends on the sign of a and the coordinates of the vertex. If a > 0, the parabola has a minimum point, and the range is all y values greater than or equal to the minimum y.
对于二次函数 f(x) = ax² + bx + c，图像是一条抛物线。值域取决于 a 的符号和顶点坐标。若 a > 0，抛物线有最低点，值域为大于等于该最小 y 值的所有 y。
, a = −2 so it has a maximum. The turning point is at (2, 3), so the range is {y : y ≤ 3}.
可以通过配方或利用对称轴 x = −b/(2a) 找到最大值或最小值。对于 f(x) = −2x² + 8x − 5，a = −2，因此有最大值。顶点为 (2, 3)，值域为 {y : y ≤ 3}。
10. Domain and Range of Inverse Functions | 反函数的定义域与值域
For a function f and its inverse f⁻¹, the domain of f⁻¹ is exactly the range of f, and the range of f⁻¹ is the domain of f. This is because an inverse swaps the input and output.
对于函数 f 和它的反函数 f⁻¹，f⁻¹ 的定义域恰好是 f 的值域，而 f⁻¹ 的值域则是 f 的定义域。这是因为反函数交换了输入和输出。
Suppose f(x) = √x − 2 ange {y : y ≥ 2}, but note that we must restrict the domain of the inverse to maintain one‑to‑one correspondence.
假设 f(x) = √x − 2，定义域为 {x : x ≥ 2}，值域为 {y : y ≥ 0}。那么它的反函数 f⁻¹(x) = x² + 2 的定义域将是 {x : x ≥ 0}，值域为 {y : y ≥ 2}，但要注意反函数可能需要限制定义域以保持一一对应。
In IGCSE problems, you might be asked to write down the domain or range of an inverse without finding the inverse explicitly, simply by swapping the domain and range of the original function.
在IGCSE题目中，你可能会被要求写出反函数的定义域或值域，而无需先求出反函数表达式，只需将原函数的定义域和值域互换即可。
11. Mapping Diagrams and Domain/Range | 映射图与定义域/值域
Mapping diagrams provide a simple visual way to see domain and range. Elements of the domain are listed in one oval, elements of the range in another, and arrows show how each input maps to an output.
映射图提供了一个简单的可视化方式来观察定义域和值域。定义域的元素列在一个椭圆中，值域的元素列在另一个椭圆中，箭头表示每个输入如何映射到输出。
For a mapping to represent a function, every element in the domain oval must have exactly one arrow leaving it. The set of all elements that are actually pointed to in the range oval forms the range.
要使一个映射表示一个函数，定义域椭圆中的每个元素必须有且只有一条出发的箭头。值域椭圆中所有实际被箭头指向的元素的集合就构成了值域。
ule f(x) = 2x. Even with limited sets, the principle of domain and range remains the same.
你可能会看到只包含几个具体数字的映射图，例如定义域 = {1, 2, 3} 通过 f(x) = 2x 映射到值域 = {2, 4, 6}。即使集合很小，定义域和值域的原理依然相同。
Mapping diagrams are especially useful when first learning the concepts, and they reinforce the idea that the range is exactly the set of outputs that the function actually achieves.
映射图在刚开始学习这些概念时特别有用，它们强调了值域就是函数实际产生的输出的集合。
12. Common Mistakes and Tips | 常见错误与技巧
One frequent mistake is confusing domain with range. Remember: domain refers to x‑values (input), range refers to y‑values (output). Reading the question carefully and labelling axes on a sketch helps avoid this.
一个常见错误是把定义域和值域搞混。记住：定义域说的是 x 值（输入），值域说的是 y 值（输出）。仔细读题并在草图上标出坐标轴有助于避免混淆。
Another common error is forgetting to write 'x ∈ ℝ' when describing a domain. Always specify that the numbers are real unless told otherwise. Also, use round brackets when listing multiple restrictions separated by commas inside set braces.
另一个常见错误是在描述定义域时忘记写“x ∈ ℝ”。除非另有说明，总要指明这些数是实数。此外，在集合花括号内用逗号分隔多个限制条件，注意不要错用符号。
en finding the range of a quadratic, students sometimes give only the maximum or minimum y‑ urning it into an inequality. Always express the range as a full set such as {y : y ≥ −4}.
求二次函数的值域时，学生有时只写出最大或最小的 y 值，却没有把它变成不等式。始终要把值域表示成一个完整的集合，例如 {y : y ≥ −4}。
With rational functions, do not cancel common factors before identifying the domain. The domain is based on the original denominator; otherwise you might miss excluded values.
处理有理函数时，不要在还没有找出定义域之前就约分。定义域是基于原始分母的，否则可能会遗漏被排除的值。
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