1.15 代数函数

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  代数函数

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  Algebraic Functions
  ::代数函数

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  A function is a special kind of relation . In a function, for each input there is exactly one output; in a relation, there can be more than one output for a given input.
  ::函数是一种特殊类型的关系。在函数中,对每个输入来说,一个输出完全是一个输出;对一个输入来说,一个输入可以有一个以上的输出。

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  Looking at a Relation 
  ::看着关系

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  Consider the relation that shows the heights of all students in a class. The domain is the set of people in the class and the range is the set of heights. This relation is a function because each person has exactly one height. If any person had more than one height, the relation would not be a function.
  ::考虑显示班级中所有学生身高的关系。 域名是班级中的一组人, 范围是一组身高。 此关系是一个函数, 因为每个人的身高都是一高度。 如果任何人身高超过一个身高, 则关系不是一个函数 。

  

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  Notice that even though the same person can’t have more than one height, it’s okay for more than one person to have the same height. In a function, more than one input can have the same output, as long as more than one output never comes from the same input.
  ::请注意,即使同一个人的身高不能超过一个高度,但一个以上的人的身高也可以相同。 在函数中,不止一个输入可以有相同的输出,只要不止一个输出从不来自同一个输入。

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  Determining if a Relation is a Function 
  ::确定关系是否为函数

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  Determine if the relation is a function.
  ::确定关系是否是一个函数。

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  The easiest way to figure out if a relation is a function is to look at all the $\begin{array}{r}x\end{array}$ -values in the list or the table. If a value of $\begin{array}{r}x\end{array}$ appears more than once, and it’s paired up with different $\begin{array}{r}y\end{array}$ -values, then the relation is not a function.
  ::确定关系是否是一个函数的最简单方法就是查看列表或表格中的所有 x 值。 如果 x 值出现不止一次, 并且与不同的 y 值相匹配, 那么关系不是一个函数 。

  1. (1, 3), (-1, -2), (3, 5), (2, 5), (3, 4)

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  You can see that in this relation there are two different $\begin{array}{r}y\end{array}$ -values paired with the $\begin{array}{r}x\end{array}$ -value of 3. This means that this relation is not a function.
  ::您可以看到,在此关系中,有两种不同的 Y 值与 X 值 3 相配。 这意味着此关系不是一个函数 。

  2. (-3, 20), (-5, 25), (-1, 5), (7, 12), (9, 2)

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  Each value of $\begin{array}{r}x\end{array}$ has exactly one $\begin{array}{r}y\end{array}$ -value. The relation is a function.
  ::x 的每个值都有一个 Y 值。 关系是一个函数 。

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  3.    $\begin{array}{r}x21012\\ y1210864\end{array}$
  ::3. x 2 1 012y1210864

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  In this relation there are two different $\begin{array}{r}y\end{array}$ -values paired with the $\begin{array}{r}x\end{array}$ -value of 2 and two $\begin{array}{r}y\end{array}$ -values paired with the $\begin{array}{r}x\end{array}$ -value of 1. The relation is not a function.
  ::在这方面,有两种不同的Y值配以2的x值和1的x值配以2的2的y值和1的x值配以2的y值。 关系不是一个函数。

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  Determining if a Graph is a Function
  ::确定图图是否为函数

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  When a relation is represented graphically, we can determine if it is a function by using the  vertical line test .  If you can draw a vertical line that crosses the graph in more than one place, then the relation is not a function.
  ::当一个关系用图形表示时,我们可以通过使用垂直线测试来确定它是否是一个函数。如果您能够绘制一条垂直线以不止一个地方跨过图形,则关系不是一个函数。

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  For the following graphs, determine whether they are functions.
  ::对于下图,确定它们是否为函数。

  

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  Not a function. It fails the vertical line test.
  ::不是函数。 它失败了垂直线测试 。

  

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  A function. No vertical line will cross more than one point on the graph.
  ::函数 A 。 任何垂直线都不会在图形上超过一个点。

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  Watch this video for help with the Examples above.
  ::观看此视频, 帮助了解上面的例子 。

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  Examples
  ::实例

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  For the following graphs, determine whether they are functions.
  ::对于下图,确定它们是否为函数。

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  Example 1
  ::例1

  

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  A function. No vertical line will cross more than one point on the graph.
  ::函数 A 。 任何垂直线都不会在图形上超过一个点。

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  Example 2
  ::例2

  

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  Not a function. It fails the vertical line test.
  ::不是函数。 它失败了垂直线测试 。

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  Review
  ::回顾

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  In 1-8, determine whether each relation is a function:
  ::在1-8中,确定每一关系是否是职能:

  1. (1, 7), (2, 7), (3, 8), (4, 8), (5, 9)
  2. (1, 1), (1, -1), (4, 2), (4, -2), (9, 3), (9, -3)
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  3.   $\begin{array}{r}\\ x\text{-}4\text{-}3\text{-}2\text{-}10\\ y1694\phantom{\text{-}}10\\ \end{array}$  
     ::x-4-3-2-10-10y169 4-10
  4. (2, -6), (1, -3), (0, 0), (1, 3), (2, 6)
  5. (-2, -8), (-1, -1), (0, 0), (1, 1), (2, 8)
  6. (-5, 10), (-1, 5), (0, 10), (1, 5), (5, 10)
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  7. $\begin{array}{r}\\ & x0\phantom{\text{-}}1101001000\\ & y2\text{-}22\phantom{\text{0}}\text{-}22\\ \end{array}$
     ::x0-110101000000y2-2 20-22
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  8. $\begin{array}{rlrlrlrlrlrl}& \text{Age}& & 20& & 25& & 25& & 30& & 35\\ & \text{Number of jobs}& & 3& & 4& & 7& & 4& & 2\\ & \text{(in the past 5 years)}\end{array}$
     ::工作数目 3 4 7 4 4 4(过去5年中)

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  In 9-10, use the vertical line test to determine whether each relation is a function.
  ::9-10时,使用垂直线测试来确定每一关系是否是一个函数。

  9. 
  10. 

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。