7.15 职能领域和范围

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  函数的域域和范围

  

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  Joanna and Macy are helping with a research project about how fish waste is affecting the local environment. They know that fish length is related to the amount of waste produced by each fish. They also know that fish in the fish farm can be anywhere from 2 inches to 30 inches. The amount of waste (in mL) per day is approximately equal to .62 times the fish length.
  ::Joanna和Mace正在帮助一项有关鱼废物如何影响当地环境的研究项目,他们知道鱼的长度与每条鱼产生的废物数量有关,他们还知道养鱼场内的鱼可能在2英寸到30英寸之间,每天的废物(毫升)约为鱼长度的0.62倍。

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  Joanna and Macy are making a chart that captures what they know. Each fish length corresponds to a volume of waste. So far they recorded fish lengths of 2.8, 11.5, 10.7, 19.3, and 29. The corresponding amounts of waste are 1.736, 7.13, 6.634, 11.966, and 17.98.
  ::Joanna和Mace正在绘制一张图表,记录他们所知道的。每条鱼的长度相当于废物的数量,到目前为止记录到的鱼的长度为2.8、11.5、10.7、19.3和29,相应的废物数量为1.736、7.13、6.634、11.966和17.98。

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  Can you calculate the total daily amount of waste in the fish farm by creating a function and stating the domain range of that function?
  ::您能否通过创建函数和说明该函数的域域范围来计算养鱼场每天的废物总量 ?

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  In this concept, you will learn to identify the of a simple linear function .
  ::在此概念中,您将学会识别一个简单的线性函数。

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  Identifying the Domain and Range of a Function
  ::确定职能的域域和范围

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  A function is a rule that maps elements of one set, the input , to elements of another set, the output . In a function, an element of the input can be mapped to only one element in the output set. This means you cannot have one input being assigned to two different outputs.
  ::函数是一个规则, 用于绘制一个集的元素、 输入、 到另一个集的元素、 输出。 在函数中, 输入的一个元素只能被映射到产出集中的一个元素。 这意味着您不能将一个输入指定给两个不同的输出 。

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  One way to represent a function is as a set of ordered pairs. The first element in the ordered pair is an element of the input and it is assigned to the second element of the ordered pair, the output.
  ::代表函数的一种方式是作为一组有顺序的配对。在有顺序的配对中,第一个元素是输入的一个元素,它被分配到有顺序的配对的第二个元素,即输出。

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  While the rule that does this assigning might not be apparent, you can still represent the function as a set of ordered pairs.
  ::虽然进行这种分配的规则可能并不明显,但您仍然可以代表一对定购夫妇的功能。

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  Let’s look at an example of a function being described by a set of ordered pairs. Notice that braces, $\begin{array}{r}\{\}\end{array}$ , are used to surround the set of ordered pairs.
  ::让我们看看一组定购配对描述的函数的例子。 注意括号 , , 用来包围一组定购配对。

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

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  In $\begin{array}{r}(1,6)\end{array}$ , 1 is an element in the input and it is being assigned to an element, 6, in the output. Each element, in the input $\begin{array}{r}(0,1,2,3)\end{array}$  is being assigned to only one element in the output $\begin{array}{r}(5,6,7,8)\end{array}$ , so this list of ordered pairs does represent a function.
  ::1,6,1是输入中的一个元素,它被指定为输出中的6,6。输入中的每个元素(0,1,2,3)仅被指定为输出中的一个元素(5,6,7,8),因此这个有顺序的配对列表确实代表一个函数。

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  Let’s take a look at another set of ordered pairs and determine if it represents a function or not and list the elements of the input and elements of the output.
  ::让我们看看另一组有顺序的配对, 以确定它是否代表函数, 并列出输入元素和输出元素 。

  $$\begin{array}{r}\{(2,4),(5,3),(6,7),(2,8)\}\end{array}$$

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  The elements of the input are $\begin{array}{r}(2,5,6)\end{array}$  and the elements of the output are $\begin{array}{r}(4,3,7,8)\end{array}$ . However this set of ordered pairs does not represent a function. It violates one of the conditions of a function. The element 2, from the input is being assigned to two different outputs, 4 and 8. This doesn’t work for functions.
  ::输入的元素是(2,5,6),输出的元素是(4,3,7,8),然而,这组定购的配对并不代表函数。它违反了函数的一个条件。输入的元素2被分配到两个不同的产出,即4和8。这对配制的配方不用于功能。

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  More formally, the domain of a function is the set of all the elements in the input and the range is the set of all the elements in the output.
  ::更正式地说,函数的域是输入中所有元素的集合,范围是输出中所有元素的组合。

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  Let’s look at another example.
  ::让我们再看看另一个例子。

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  Identify the domain and range of this function (or a rule that assigns an element of the input to an element of the output) represented as the following set of ordered pairs.
  ::指定该函数的域和范围(或为输出要素指定输入元素的规则),作为以下一组有顺序的对子。

  $$\begin{array}{r}\{(0,-10),(2,-8),(4,-6),(6,-4)\}\end{array}$$

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  The domain is the set of all the elements in the input, or all of the first elements of the ordered pairs.
  ::域是输入中所有元素的集合, 或定购对子的所有初始元素 。

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  The answer is the domain of this function is $\begin{array}{r}\{0,2,4,6\}\end{array}$ .
  ::答案是此函数的域为 {0,2,4,6}。

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  The range is the set of all the elements in the output, or all of the second elements in the ordered pair.
  ::范围是输出中所有元素的集合,或定购对中所有第二元素的组合。

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  The answer is the range of this function is $\begin{array}{r}\{-10,-8,-6,-4\}\end{array}$ .
  ::答案是此函数的范围为 10, - 8, - 6 - 4}。

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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 Joanna and Macy, who are researching how waste from local fish farms is impacting the environment.
  ::早些时候,你曾遇到乔安娜和梅西的问题, 他们正在研究当地养鱼场的废物对环境的影响。

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  They have recorded fish lengths of 2.8, 11.5, 10.7, 19.3, and 29. The corresponding amounts of waste are 1.736, 7.13, 6.634, 11.966, and 17.98.
  ::它们记录的鱼类长度分别为2.8、11.5、10.7、19.3和29,相应的废物量为1.736、7.13、6.634、11.966和17.98。

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  This information can be represented as a set or ordered pairs.
  ::这种信息可以作为一对或一对或一对定购的一对来表示。

  $$\begin{array}{r}\{(2.8,1.736),(11.5,7.13),(10.7,6.634),(19.3,11.966),(29,17.98)\}\end{array}$$

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  The five entries Joanna and Macy recorded represent a function with the input being the fish length and the output being amount of waste.
  ::记录到的5个条目Joanna和Mace是一个功能,投入是鱼的长度,产出是废物的数量。

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  The domain of this function is the fish lengths, or $\begin{array}{r}\{2.8,11.5,10.7,19.3,29\}\end{array}$ , and the range is the waste amount in mL, or $\begin{array}{r}\{1.736,7.13,6.634,11.966,17.98\}\end{array}$ .
  ::这个功能的域是鱼的长度,或 {2.8,11.5,10.7,19.3,29},其范围是 毫升的废物量,或 {1.736,7.13,6.6,634,11.966,17.98}。

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  Example 2
  ::例2

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  Write the domain and range of this function, represented as a set of ordered pairs.
  ::写入此函数的域和范围, 以一组有顺序的配对表示 。

  $$\begin{array}{r}\{(1,3)(3,9)(4,6)(5,12)\}\end{array}$$

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  The first value of each ordered pair is the domain of the function.
  ::每对定购的一对的第一个值是函数的域 。

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  The second value of each ordered pair is the range of the function.
  ::每对定购的对的第二个值是函数的范围。

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  The answer is the domain is $\begin{array}{r}\{1,3,4,5\}\end{array}$  and the range is $\begin{array}{r}\{3,9,6,12\}\end{array}$ .
  ::答案是域是 {1,3,4,5},范围是 {3,9,6,12}。

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  Identify the domain and range of each function, represented as a set of ordered pairs.
  ::确定每个函数的域和范围,作为一组有顺序的对子。

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  Example 3
  ::例3

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

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  The first value of each ordered pair is the domain of the function.
  ::每对定购的一对的第一个值是函数的域 。

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  The second value of each ordered pair is the range of the function.
  ::每对定购的对的第二个值是函数的范围。

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  The domain is $\begin{array}{r}\{1,2,5,9\}\end{array}$  and the range is $\begin{array}{r}\{3,4,7,11\}\end{array}$ .
  ::域为 {1,2,5,9},范围为 {3,4,7,11}。

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  Example 4
  ::例4

  $$\begin{array}{r}\{(8,12)(9,22)(4,7)(2,5)\}\end{array}$$

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  The first value of each ordered pair is the domain of the function.
  ::每对定购的一对的第一个值是函数的域 。

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  The second value of each ordered pair is the range of the function.
  ::每对定购的对的第二个值是函数的范围。

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  The domain is $\begin{array}{r}\{8,9,4,2\}\end{array}$ , and the range is $\begin{array}{r}\{12,22,7,5\}\end{array}$ .
  ::域是 {8,9,4,2},范围是 {12,22,7,5}。

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  Example 5
  ::例5

  $$\begin{array}{r}\{(8,9)(3,5)(7,6)(10,12)\}\end{array}$$

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  The first value of each ordered pair is the domain of the function.
  ::每对定购的一对的第一个值是函数的域 。

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  The second value of each ordered pair is the range of the function.
  ::每对定购的对的第二个值是函数的范围。

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  The domain is $\begin{array}{r}\{8,3,7,10\}\end{array}$ , and the range is $\begin{array}{r}\{9,5,6,12\}\end{array}$ .
  ::域是 {8,3,7,10},范围是 {9,5,6,12}。

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  Review
  ::回顾

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  Identify whether the set of ordered pairs could represent a function or not.
  ::确定一组定购配对是否代表一种功能。

  1. $\begin{array}{r}\{(1,3)(2,6)(2,5)(3,7)\}\end{array}$  
  2. $\begin{array}{r}\{(2,5)(3,6)(4,7)(5,8)\}\end{array}$  
  3. $\begin{array}{r}\{(6,1)(7,2)(8,3)\}\end{array}$  
  4. $\begin{array}{r}\{(5,2)(5,3)(5,4)(5,5)\}\end{array}$  
  5. $\begin{array}{r}\{(81,19)(75,18)(76,18)(77,19)\}\end{array}$  

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  For problems 6-10, state the domain of the function in numbers 1-5.
  ::对于问题6-10,用数字1-5说明该函数的范围。

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  For problems 11-15, state the range of the function in numbers 1-5.
  ::对于问题11-15, 请用数字 1-5 表示函数范围 。

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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 。