📚 Domain and Range | 定义域与值域
In IGCSE Edexcel Mathematics, understanding the domain and range of a function is essential for sketching graphs, solving equations, and working with composite functions. The domain of a function is the complete set of possible input values, while the range is the complete set of possible output values produced by the function.
在爱德思IGCSE数学中，理解函数的定义域和值域对于绘制函数图像、解方程以及处理复合函数至关重要。函数的定义域是所有可能输入值的集合，而值域则是函数产生的所有可能输出值的集合。
1. Definition of Domain and Range | 定义域与值域的定义
For a function f(x), the domain is the set of all possible values of x for which the function is defined. The range is the set of all possible values of f(x) (or y) that result from substituting every value in the domain into the function.
对于函数f(x)，定义域是使函数有意义的所有x值的集合。值域是将定义域中的每一个值代入函数后得到的所有f(x)（即y）值的集合。
Think of a function as a machine: you feed it inputs (x), and it produces outputs (y). The domain tells you what you are allowed to put into the machine; the range tells you everything that can come out of the machine.
可以把函数想象成一台机器：你输入自变量x，它产生因变量y。定义域告诉你允许输入什么，值域告诉你机器可能输出什么。
2. Set Notation and Interval Notation | 集合记号与区间记号
At IGCSE level, you will encounter two main ways to write domains and ranges: set notation and interval notation.
在IGCSE阶段，你会遇到两种主要的写法：集合记号和区间记号。
uses braces and conditions. For example, the domain of f(x) = 1/x can be written as {x : x ∈ ℝ, x ≠ 0}. This is read as "the set of x such that x is a real number and x is not equal to zero."
集合记号使用大括号和条件。例如，f(x) = 1/x的定义域可以写成{x : x ∈ ℝ, x ≠ 0}，读作"所有满足x是实数且x不等于0的x的集合"。
bracket [ means the endpoint is included; a round parenthesis ( means the endpoint is excluded.
区间记号使用方括号和圆括号。例如，定义域x ≥ 2可以写成[2, ∞)。方括号 [ 表示包含端点；圆括号 ( 表示不包含端点。
| Notation | Meaning | Example |
| {x : condition} | Set of x satisfying condition | {x : x > 3} |
| [a, b] | a ≤ x ≤ b | [1, 5] |
| (a, b) | a < x < b | (1, 5) |
| [a, ∞) | x ≥ a | [2, ∞) |
3. Domain and Range of Linear Functions | 线性函数的定义域与值域
x + c, where m is the gradient and c is the y-intercept. For any linear function with m ≠ 0, the domain is all real numbers and the range is also all real numbers.
线性函数的形式为f(x) = mx + c，其中m是斜率，c是y轴
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