9.1 职能

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  承认职能职能

  

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  Tammy has opened an ice cream stand to earn money for her college tuition. She records the number of cones she sells each day in a table like the one shown below. 
  ::塔米开了一个冰淇淋摊来挣学费,她每天在一张像下面的桌子上 记录她卖出的甜筒的数量。

  Days in operation	Number of cones sold
  1	48
  2	56
  3	77
  4	79
  5	82
  6	98
  7	105

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  Tammy is very pleased with the numbers of cones sold for her first week in operation. Now she wants to analyze the results shown in the table. What information can Tammy record from her table?
  ::现在她想分析表格中显示的结果。

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  In this concept, you will learn to recognize functions.
  ::在这个概念中,你会学会识别功能。

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  Functions
  ::职能职能职能职能职能职能职能职能职能职能职能职能职能职能职能

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  Many things in the real world consist of a relationship between two things. The number of hours a person works and the amount of money earned, the number of gallons of gas bought and the distance a person can drive and the names of the students in a class and their heights are just a few examples of relationships between two things.
  ::在现实世界中,许多事情都包含着两种事物之间的关系。 一个人工作的时间和挣得的钱的数量、购买的天然气加仑的数量和一个人能开车的距离、一个班的学生的姓名和他们的身高只是两个事物之间关系的几个例子。

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  The names of the students in a class paired with their heights is an example of a relation . These two things can be paired as (name, height) or as (height, name). The name and the height are written as an ordered pair which simply means that one comes first and the other comes second. When a relationship is expressed as an ordered pair it is called a relation . A set of ordered pairs is a relation .
  ::与身高相配的班级学生的姓名是关系的一个实例。这两件事可以配对为(姓名、身高)或(身高、姓名),姓名和身高是按顺序排列的一对书写的,这只意味着一人先来,另一人次来。当关系以定顺序排列的一对表示时,它被称为关系。一对定顺序的一对是关系。一对定顺序的一对是关系。

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  If the relationship between the names of the students and their heights is written as (name, height) with the name as the first thing in the ordered pair, then the names are the set of all staring points. This set of starting points is called the domain of the relation. The heights are the second thing in the ordered pair and are the set of all ending points. This set of ending points is called the range of the relation.
  ::如果学生姓名和身高之间的关系以(名称、身高)写成,且该名称是定购一对中的第一个名字,那么名称就是所有凝视点的一组。这组起始点被称为关系领域。高度是定购一对中的第二个东西,是所有结束点的一组。这组结束点被称为关系的范围。

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  If the variable ‘ $\begin{array}{r}x\end{array}$ ’ represents the names of the students and the variable ‘ $\begin{array}{r}y\end{array}$ ’  represents the heights of the students, then the ordered pair (name, height) can be written as $\begin{array}{r}(x,y)\end{array}$ . The set of values for ‘ $\begin{array}{r}x\end{array}$ ’ will be the domain and the set of values for ‘ $\begin{array}{r}y\end{array}$ ’ will be the range.
  ::如果变量 'x ' 代表学生的姓名,变量 'y ' 代表学生的高度,那么订购的一对(名称、身高)可以写成(x,y) 。 “x”的一组值将是域名,“y”的一组值将是范围。

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  Look at the following set of ordered pairs:
  ::看看下面这组定购的一对:

  $$\begin{array}{r}(1,4),(2,6),(2,8),(3,10),(4,12)\end{array}$$

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  The values are written as ordered pairs $\begin{array}{r}(x,y)\end{array}$  such that the domain includes the values $\begin{array}{r}(1,2,3,4)\end{array}$  and the range includes the values $\begin{array}{r}(4,6,8,10,12)\end{array}$ . Notice that the $\begin{array}{r}x\end{array}$ -value of 2 is paired with two different $\begin{array}{r}y\end{array}$ -values. A relation can have a member of the domain paired with more than one member from the range.
  ::数值是按顺序排列的一对(x,y)写成,以便域包括值(1,2,3,4),范围包括值(4,6,8,10,12)。请注意,2的X值与两个不同的 Y值相配。关系中可以有一个域成员与一个以上的域成员对配。

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  Look at this set of ordered pairs:
  ::看看这组订购的一对:

  $$\begin{array}{r}(1,3),(2,5),(3,7),(4,9),(5,11)\end{array}$$

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  The values are written as ordered pairs $\begin{array}{r}(x,y)\end{array}$  such that the domain includes the values $\begin{array}{r}(1,2,3,4,5)\end{array}$  and the range includes the values $\begin{array}{r}(3,5,7,9,11)\end{array}$ . Notice that each $\begin{array}{r}x\end{array}$ -value is paired with only one $\begin{array}{r}y\end{array}$ -value. A relation such that each member of the domain is paired with one and only one member of the range is called a function .
  ::数值是按顺序排列的一对(x,y)写成,以便域包括值(1,2,3,4,4,5),范围包括值(3,5,7,9,11)。请注意,每一x值只配一个y值。域内每个成员配上一个成员,而区域内只有一个成员,这种关系被称为函数。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about  Tammy and her ice cream cones. She wants to record information from the table.
  ::早些时候,你听说塔米和她的冰淇淋锥出了问题 她想从桌子上录下信息

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  Tammy can record the from the table and determine if the information represents a relation or a function.
  ::塔米可以从表格中记录信息并确定信息是关系还是函数。

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  First, make a list of ordered pairs to represent the information in the table.
  ::首先,列出一份有顺序的对子名单,以代表表格中的信息。

  $$\begin{array}{r}\{(1,48),(2,56),(3,77),(4,79),(5,82),(6,98),(7,105)\}\end{array}$$

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  Next, list the members of the domain. Remember to list each $\begin{array}{r}x\end{array}$ -value only once.
  ::下一步请列出域的成员。 记住只列出一次每个 x 值 。

  $$\begin{array}{r}(1,2,3,4,5,6,7)\end{array}$$

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  Next, list the members of the range. Remember to list each $\begin{array}{r}y\end{array}$ -value only once.
  ::下一步, 列出范围的成员。 记住只列出一次 Y 值 。

  $$\begin{array}{r}(48,56,77,79,82,98,105)\end{array}$$

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  Next, look at the ordered pairs to see if any $\begin{array}{r}x\end{array}$ -value is paired with more than one $\begin{array}{r}y\end{array}$ -value.
  ::接下来,看看订购的一对,看看有没有一个以上的 Y 值的X值配对。

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  Every $\begin{array}{r}x\end{array}$ -value from the domain is paired with only one of the $\begin{array}{r}y\end{array}$ -values from the range.
  ::域中的每一 x 值都与范围中的一个 Y 值相匹配。

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  Then, state if the ordered pairs represent a relation or a function.
  ::然后,说明定购的对子是否代表一种关系或一种功能。

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  A function.
  ::A 函数。

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  The answer is a function.
  ::答案是一个函数。

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  Example 2
  ::例2

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  Determine if the following represents a function or a relation.
  ::确定以下是函数还是关系。

  

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  First, follow each arrow from its value in the domain to its value in the range.
  ::首先,跟随每一箭头,从它在域内的值到它在射程中的值。

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  Next, write a set of ordered pairs to represent the arrows. Remember the domain represents the $\begin{array}{r}x\end{array}$ -values and the range represents the $\begin{array}{r}y\end{array}$ -values.
  ::下一步, 写入一组有顺序的对以代表箭头。 记住, 域代表 x 值, 区域代表 Y 值 。

  $$\begin{array}{r}(-3,4),(-2,8),(-1,0),(0,8),(1,-4)\end{array}$$

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  Next, check to see if each $\begin{array}{r}x\end{array}$ -value is paired with one $\begin{array}{r}y\end{array}$ -value.
  ::下一步,检查是否每个 x 值配以一个 Y 值 。

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  Then, state if the ordered pairs represent a relation or a function.
  ::然后,说明定购的对子是否代表一种关系或一种功能。

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  A function.
  ::A 函数。

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  The answer is a function.
  ::答案是一个函数。

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  Example 3
  ::例3

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  Do the following ordered pairs represent a relation or a function?
  ::下列有顺序的配对是代表一种关系还是函数?

  $$\begin{array}{r}\{(0,5),(1,6),(2,7),(1,8),(3,9),(2,10)\}\end{array}$$

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  First, list the members of the domain. Remember to list each $\begin{array}{r}x\end{array}$ -value only once.
  ::首先列出域的成员。 记住只列出一次每个 x 值 。

  $$\begin{array}{r}(0,1,2,3)\end{array}$$

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  Next, list the members of the domain. Remember to list each $\begin{array}{r}y\end{array}$ -value only once.
  ::下一步, 列出域的成员。 记住只列出一次 Y 值 。

  $$\begin{array}{r}(5,6,7,8,9,10)\end{array}$$

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  Next, look at the ordered pairs to see if any $\begin{array}{r}x\end{array}$ -value is paired with more than one $\begin{array}{r}y\end{array}$ -value.
  ::接下来,看看订购的一对,看看有没有一个以上的 Y 值的X值配对。

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  $\begin{array}{r}(1,6)\end{array}$  and $\begin{array}{r}(1,8)\end{array}$ . The $\begin{array}{r}x\end{array}$ -value of 1 is paired with two different $\begin{array}{r}y\end{array}$ -values 6 and 8.
  :1,6)和(1,8),1的x值加上两个不同的y值6和8。

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  $\begin{array}{r}(2,7)\end{array}$  and $\begin{array}{r}(2,10)\end{array}$ . The $\begin{array}{r}x\end{array}$ -value of 2 is paired with two different $\begin{array}{r}y\end{array}$ -values 7 and 10.
  :2,7)和(2,10)x值为2,乘以两个不同的y值7和10。

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  Then, state if the ordered pairs represent a relation or a function.
  ::然后,说明定购的对子是否代表一种关系或一种功能。

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  A relation.
  ::一种关系。

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  The answer is a relation.
  ::答案是一个关系。

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  Example 4
  ::例4

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  Do the following ordered pairs represent a relation or a function?
  ::下列有顺序的配对是代表一种关系还是函数?

  $$\begin{array}{r}\{(0,5),(1,5),(2,5),(3,5)\}\end{array}$$

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  First, list the members of the domain. Remember to list each $\begin{array}{r}x\end{array}$ -value only once.
  ::首先列出域的成员。 记住只列出一次每个 x 值 。

  $$\begin{array}{r}(0,1,2,3)\end{array}$$

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  Next, list the members of the domain. Remember to list each $\begin{array}{r}y\end{array}$ -value only once.
  ::下一步, 列出域的成员。 记住只列出一次 Y 值 。

  $$\begin{array}{r}(5)\end{array}$$

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  Next, look at the ordered pairs to see if any $\begin{array}{r}x\end{array}$ -value is paired with more than one $\begin{array}{r}y\end{array}$ -value.
  ::接下来,看看订购的一对,看看有没有一个以上的 Y 值的X值配对。

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  Each $\begin{array}{r}x\end{array}$ -value is paired with only one $\begin{array}{r}y\end{array}$ -value.
  ::每个X值配对只有一个y值。

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  Then, state if the ordered pairs represent a relation or a function.
  ::然后,说明定购的对子是否代表一种关系或一种功能。

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  A function.
  ::A 函数。

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  The answer is a function.
  ::答案是一个函数。

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  Example 5
  ::例5

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  Do the following ordered pairs represent a relation or a function?
  ::下列有顺序的配对是代表一种关系还是函数?

  $$\begin{array}{r}\{(8,-2),(5,-3),(0,-9),(8,-4)\}\end{array}$$

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  First, list the members of the domain. Remember to list each $\begin{array}{r}x\end{array}$ -value only once.
  ::首先列出域的成员。 记住只列出一次每个 x 值 。

  $$\begin{array}{r}(0,5,8)\end{array}$$

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  Next, list the members of the domain. Remember to list each $\begin{array}{r}y\end{array}$ -value only once. 
  ::下一步, 列出域的成员。 记住只列出一次 Y 值 。

  $$\begin{array}{r}(-9,-4,-3,-2)\end{array}$$

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  Next, look at the ordered pairs to see if any $\begin{array}{r}x\end{array}$ -value is paired with more than one $\begin{array}{r}y\end{array}$ -value.
  ::接下来,看看订购的一对,看看有没有一个以上的 Y 值的X值配对。

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  The $\begin{array}{r}x\end{array}$ -value of 8 is paired with two $\begin{array}{r}y\end{array}$ -values -2 and -4.
  ::8的x值加上两个y值-2和-4。

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  Then, state if the ordered pairs represent a relation or a function.
  ::然后,说明定购的对子是否代表一种关系或一种功能。

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  A relation.
  ::一种关系。

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  The answer is a relation.
  ::答案是一个关系。

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  Review
  ::回顾

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  Are the relations functions? Write function if it is a function and not a function if it is not a function.
  ::是关系函数吗? 写入函数如果是一个函数,而不是函数如果不是函数,则写函数。

  1. $\begin{array}{r}\{(4,7),(8,11),(4,9),(8,13)\}\end{array}$
  2. $\begin{array}{r}\{(2,7),(2,11),(4,12),(8,13)\}\end{array}$
  3. $\begin{array}{r}\{(3,4),(5,6),(7,8),(8,10)\}\end{array}$
  4. $\begin{array}{r}\{(12,7),(11,11),(14,9),(18,13)\}\end{array}$
  5. $\begin{array}{r}\{(3,7),(4,11),(3,9),(12,13)\}\end{array}$
  6. $\begin{array}{r}\{(8,7),(9,6),(10,5),(11,4)\}\end{array}$
  7. $\begin{array}{r}\{(4,2),(8,1),(3,9),(8,7)\}\end{array}$
  8. $\begin{array}{r}\{(11,17),(18,21),(14,19),(18,13)\}\end{array}$
  9. $\begin{array}{r}\{(4,7),(8,11),(4,9),(8,13)\}\end{array}$
  10. $\begin{array}{r}\{(-3,0),(-2,0),(-1,0),(0,0),(1,1)\}\end{array}$
  11. $\begin{array}{r}\{(6,25),(12,35),(18,45),(24,55)\}\end{array}$
  12. $\begin{array}{r}\{(2,4),(3,5)(2,6),(7,9)\}\end{array}$

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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 。