senior high maths [Ch1.] transformations ⅢFunctions 北美高中数学第一章 转换 函数

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

The next concern is to find out what kind of relations are functions. First, a function is defined as follows:"For every value of the domain(x-value) there is one and only one value for the range(y-value)." What does this mean? It says any x-value can only have one y value. For example:
下一項重要的事是找出函數是怎樣的關係。首先，函數可以定義為：“對於定義域內每一個ｘ值，有且只有值域內的一個ｙ值與其相對應。” 這是什麼意思呢？它是說每個ｘ值只能有一個ｙ值。如以下例子：

a) (1,3), (2,4), (3,-1) is a function because each x-value 1, 2, 3 has only one value for y.  
　　(1,3), (2,4), (3,-1)是函數因為每個ｘ值１，２，３只有一個ｙ值。
b) (1,3), (-1,3), (2,4)is a function.
 　　 (1,3), (-1,3), (2,4)是函數。
c) (1,3), (1,2), (4,5) is not a function because x=1 gives y=2 and y=3, i.e., two values.
　　(1,3), (1,2), (4,5)不是函數因為ｘ＝１時，ｙ＝２並且ｙ＝３，有兩個值。

If "y" is an even power, then it can't be a function, e.g.,

X²+y²=9, y=±√(9-x²), (two values of y for each x value). If you can graph the equation, then remember the vertical line test: any vertical line can only cross the graph once if it is a function.
如果ｙ是偶次方的話，那麼他不可能成為函數，例如X²+y²=9, y=±√(9-x²)（有兩個ｙ值對應每個ｘ值）。如果你可以畫出函數圖像的話，記得垂直線測試：如果他是函數，任何一條垂直線只能與其圖形有一個交點。

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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

senior high maths [Ch1.] transformations Ⅰbasic formulas 北美高中数学第一章 转换 基本公式

Basic formulas基本公式

1. 0⁰ =undefined 无法运算

2. x²· x³=x²⁺³=x⁵ 

3. 2¹²/2⁴ =2^12-4=2⁸ 

4. 2^x+2^y cannot be simplified 无法简化

5. (x²)³=x⁶ 

6. (-2)⁰=1

7. x^-3=1/x³ 

8. √4=2

9. x²=4, x=±2

10. x³=8, x=2

11. x³=-8, x=-2

12. -3²=-9

13. (x+y)^-1=1/(x+y)

14. √(x+y)cannot be simplified 无法简化

15. √(9+16)=√25=5

16. (x^-1+y^-1)^-1=(1/x+1/y)^-1=((y+x)/xy)^-1=xy/(x+y)

17. √[(x-2)²]=|x-2|

18. (√x-1)²=x-2√x+1

19. x²+y²=(x-yi)(x+yi)

20. x³-y³=(x-y)(x²+xy+y²)

21. (x+2)²=x²+4x+4

22. x³+8=(x+2)(x²-2x+4)

23. x-1=(√x-1)(√x+1)

24. x²=x, x²-x=0, x(x-1)=0, x=0 or x=1

25. –x² when x=-2, -(-2)²=-4

26. 2³=8

27. 3²=9