senior high maths [Ch1.] transformations ⅡRelations 北美高中数学第一章 转换 函数关系

From Theory and problems for senior high math
来自 Theory and problems for senior high math

Analysing Functions and Relations
函数与函数关系的分析
In Mathematics, determining the relationship between two variables is a very important concept. This chapter will show how to visualise relations and functions by means of a graph. Graphs will become central to each aspect of this chapter, for both theoretical understanding and problem solving.
在数学中，确定两个变量的关系是很重要的概念。本章会教您如何使用函数图像来确定变量的关系及函数。函数图像将会成为本章各部分（理论，理解和解题）的重点内容。

1. Relations变量关系
    A relation is a set of ordered pairs, in other words, just a number of points in a coordinate plane.
变量关系是一组排列的变量数对，用其他的话说， 就是一群点的坐标。




a)

 b)

 c)

d)

Presented above are examples of relations. The set of x-values of all the points is called the domain of the relation, and the set of y-values is called the range. In example a) above, the domain is -1,1,2, and the range is -1,0,2.
以上为一些变量关系的例子。所有点的x值的集合叫做定义域，而所有点的y值的集合叫做值域。在例a）中，定义域为 -1，1，2，而值域为-1，0，2.

If you are allowed to use any set of numbers you want for the domain or range, the "all real numbers" is the answer. (See Example 1 below)
如果你可以使用任何你想要的数集来作为定义域和值域，那么可以用“所有实数”作为回答。（见例一）

Example 1:X=y²
The domain is x ≥ 0  定义域为x ≥ 0
The range is "all real numbers" 值域为“所有实数”

The domain and range become harder to identify when there is no graph. Remember, the domain is the set of x numbers, and the range is the set of y numbers used in the equation. There are two main concerns when working with domain and range in any equation:
如果没有图的话，定义域和值域会比较难找到。记得，定义域是式子中所有x的值，值域是式子中所有y的值。在解决定义域和值域问题时，有两个要点：

• not having a negative number inside an even root, and
• 偶次根号内不能有负数

• not having zero in the denominator. 
• 分母不能为0

Example 2  :y = 2-√(3-x)           
In this example, the domain is x ≤ 3 because an even root must be ≥ 0, so 3-x ≥0, x ≤ 3.
在此例中，定义域为 x ≤ 3 因为偶次根号内要 ≥ 0, 所以 3-x ≥0, x ≤ 3。
The range is y ≤ 2 because it is 2 minus the positive value of √(3-x) 
值域为 y ≤ 2 因为是2减去√(3-x) 的正值。


Translated by Annabelle Wu
翻译自  Annabelle Wu