1.11 职能的应用

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• 选择节常规

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• 选择节职能的运用

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  职能的运用

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  Using Functions to Describe a Situation
  ::使用函数描述状况

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  In many situations, we collect data by conducting a survey or an experiment, and then organize the data in a table of values. Most often, we want to find the function rule or formula that fits the set of values in the table, so we can use the rule to predict what could happen for values that are not in the table.
  ::在许多情形中,我们通过调查或实验收集数据,然后将数据组织在一个数值表中。 通常,我们想找到适合表格中一组数值的函数规则或公式,这样我们就可以使用这一规则来预测不在表中的数值会发生什么。

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  Writing Functions 
  ::写入函数

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  1. Write a function rule for the following table:
  ::1. 为下表写一条函数规则:

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  $$\begin{array}{rl}& \text{Number of CDs}246810\\ & \text{Cost in}\mathrm{\$}24487296120\end{array}$$

  
  ::CDs246810 成本数,以24 48 72 96 120美元计算

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  You pay $24 for 2 CDs, $48 for 4 CDs, $120 for 10 CDs. That means that each CD costs $12.
  ::你为2张CD支付24美元,为4张CD支付48美元,为10张CD支付120美元,这意味着每张CD费用为12美元。

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  We can write a function rule:
  ::我们可以写一个函数规则:

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  Cost $\begin{array}{r}=\mathrm{\$}12\times \end{array}$ (number of CDs) or $\begin{array}{r}f(x)=12x\end{array}$
  ::成本=12美元x(CD数)或f(x)=12x

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  2. Write a function rule for the following table:
  ::2. 为下表写一条函数规则:

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  $$\begin{array}{rl}& x-3-2-10123\\ & y3210123\end{array}$$

  
  ::x-3-2-10123y3 2 10123

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  You can see that a negative number turns into the same number, only positive, while a non-negative number stays the same. This means that the function being used here is the absolute value function: $\begin{array}{r}f(x)=\mid x\mid \end{array}$ .
  ::您可以看到负数变为相同数字, 只有正数, 而非负数则保持不变。 这意味着这里使用的函数是绝对值函数 : f( x)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\可以\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  Coming up with a function based on a set of values really is as tricky as it looks. There’s no rule that will tell you the function every time, so you just have to think of all the types of functions you know and guess which one might be a good fit, and then check if your guess is right. In this book, though, we’ll stick to writing functions for linear relationships, which are the simplest type of function.
  ::以一组数值为基础的函数的出现,真的和看上去的一样困难。没有一条规则可以告诉你每一次的函数,所以你只需要想想你所知道的所有类型的函数,猜猜哪个功能可能适合,然后检查你的猜测是否正确。在这本书中,我们会坚持为线性关系写入函数,因为线性关系是最简单的功能类型。

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  Represent a Real-World Situation with a Function
  ::a 具有职能的 " 真实世界形势 " 。

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  Let’s look at a few real-world situations that can be represented by a function.
  ::让我们看看几个可以用功能来代表的现实世界的情况。

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  1. Maya has an internet service that currently has a monthly access fee of $11.95 and a connection fee of $0.50 per hour. Represent her monthly cost as a function of connection time.
  ::1. 玛雅拥有因特网服务,目前每月上网费为11.95美元,每小时连接费为0.50美元,按连接时间计算每月费用。

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  Define
  ::定义定义

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  Let $\begin{array}{r}x=\end{array}$ the number of hours Maya spends on the internet in one month
  ::让我们 x = 一个月后玛雅在互联网上花费的小时数

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  Let $\begin{array}{r}y=\end{array}$ Maya’s monthly cost
  ::Maya的月费

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  Translate
  ::翻译

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  The cost has two parts: the one-time fee of $11.95 and the per-hour charge of $0.50. So the total cost is the flat fee $\begin{array}{r}+\end{array}$ the charge per hour $\begin{array}{r}\times \end{array}$ the number of hours.
  ::费用分为两部分:一次性收费11.95美元和每小时收费0.50美元。总费用是固定收费+每小时收费x小时数。

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  Answer
  ::答复答复答复

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  The function is $\begin{array}{r}y=f(x)=11.95+0.50x\end{array}$ .
  ::函数是 y=f(x)= 11.95+0.50x。

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  2. Alfredo wants a deck build around his pool. The dimensions of the pool are $\begin{array}{r}12feet\times 24feet\end{array}$ and the decking costs $3 per square foot. Write the cost of the deck as a function of the width of the deck.
  ::2. Alfredo想要在泳池周围建一个甲板,泳池的尺寸为12英尺×24英尺,每平方英尺的甲板费用为3美元。

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  Solution
  ::解决方案

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  Define
  ::定义定义

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  Let $\begin{array}{r}x=\end{array}$ width of the deck
  ::Let x= 甲板宽度

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  Let $\begin{array}{r}y=\end{array}$ cost of the deck
  ::让 Y = 甲板成本

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  Make a sketch and label it
  ::做个草图和贴标签

  

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  Translate
  ::翻译

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  You can look at the decking as being formed by several rectangles and squares. We can find the areas of all the separate pieces and add them together:
  ::您可以将甲板看成由多个矩形和方形组成。 我们可以找到所有分开的片段区域, 并把它们加在一起 :

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  $$\begin{array}{rl}\text{Area}& =12x+12x+24x+24x+x^2+x^2+x^2+x^2\\ & =72x+4x^2\end{array}$$

  
  ::区域=12x+12x+12x+24x+24x+x2+x2+x2+x2+x2=72x+4x2

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  To find the total cost, we then multiply the area by the cost per square foot ($3).
  ::为了找到总成本,我们再将面积乘以每平方英尺的成本(3美元)。

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  Answer
  ::答复答复答复

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  $$\begin{array}{r}f(x)=3(72x+4x^2)=216x+12x^2\end{array}$$

  
  ::f( x) = 3( 72x+4x2) = 216x+12x2

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  Example
  ::示例示例示例示例

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  Example 1
  ::例1

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  A cell phone company sells two million phones in their first year of business. The number of phones they sell doubles each year. Write a function that gives the number of phones that are sold per year as a function of how old the company is.
  ::一个手机公司在营业的第一年出售200万部手机。 他们每年出售的电话数量翻了一番。 写入一个函数, 显示每年出售的电话数量, 以公司年老为准 。

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  Define
  ::定义定义

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  Let $\begin{array}{r}x=\end{array}$ age of company in years
  ::Lex=公司年龄(年数)

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  Let $\begin{array}{r}y=\end{array}$ number of phones that are sold per year
  ::y=每年售出的电话数

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  Make a table
  ::做一个表格

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  $$\begin{array}{rl}& \text{Age (years)}1234567\\ & \text{Millions of phones}248163264128\end{array}$$

  
  ::年龄(岁)1234 5 6 7 7百万电话248163264128

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  Write a function rule
  ::写入函数规则

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  The number of phones sold per year doubles every year, so the first year the company sells 2 million phones, the next year it sells $\begin{array}{r}2\times 2\end{array}$ million, the next year it sells $\begin{array}{r}2\times 2\times 2\end{array}$ million, and so on. You might remember that when we multiply a number by itself several times we can use exponential notation: $\begin{array}{r}2=2^1,2\times 2=2^2,2\times 2\times 2=2^3,\end{array}$ and so on. In this problem, the exponent just happens to match the company’s age in years, which makes our function easy to describe.
  ::每年售出的电话数量每年翻一番,因此,公司的第一年售出200万部电话,下一年售出2x2百万台电话,下一年售出2x2百万台电话,第二年售出2x2百万台电话,等等。你可能记得,当我们自己乘以数数数数数数数数的时候,我们可以使用指数标记:2=21,2x2=22,2x2=23,等等。 在这个问题中,出价刚好和公司年岁数相符,这使我们的功能容易描述。

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  Answer
  ::答复答复答复

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  $$\begin{array}{r}y=f(x)=2^x\end{array}$$

  
  ::y=f(x)=2x

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  Review 
  ::回顾

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  1. Write a function rule for the following table:
     $$\begin{array}{r}x3456\\ y9162536\end{array}$$
     
     ::为下表写入函数规则: x34 5 6Y9162536
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  2. Write a function rule for the following table:
     $$\begin{array}{r}\text{Hours}0123\\ \text{Cost}15202530\end{array}$$
     
     ::写下表格的函数规则: Hours0 1 2 3Cost 15202530
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  3. Write a function rule for the following table:
     $$\begin{array}{r}x0123\\ y241263\end{array}$$
     
     ::为下表写入函数规则: x0 1 23y241263
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  4. Write a function that represents the number of cuts you need to cut a ribbon into $\begin{array}{r}x\end{array}$ pieces.
     ::写一个函数,该函数代表您将丝带切成 x 块时需要剪切的次数 。
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  5. Write a function that represents the number of cuts you need to divide a pizza into $\begin{array}{r}x\end{array}$ slices.
     ::写一个函数, 表示您需要将比萨饼分成 x 切片的切分数 。

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  For 6-8, suppose Solomon charges a $40 flat rate plus $25 per hour to repair a leaky pipe.
  ::6-8,假设所罗门 收费40美元 统一费率加每小时25美元 来修理漏水管道

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  6. Write a function that represents the total fee charged as a function of hours worked.
     ::写作功能代表按工作小时计算的总收费。
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  7. How much does Solomon earn for a 3-hour job?
     ::所罗门3小时工作挣多少钱?
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  8. How much does he earn for three separate 1-hour jobs?
     ::他三次单干一个小时的工作能挣多少钱?

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  For 9-12, suppose Rochelle has invested $2500 in a jewelry making kit. She makes bracelets that she can sell for $12.50 each.
  ::假设Rochelle投资了2500美元给一个珠宝制作工具箱。她制作手镯,每件12.50美元。

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  9. Write a function that shows how much money Rochelle makes from selling $\begin{array}{r}b\end{array}$ bracelets.
     ::写一个函数,显示Rochelle从销售b 手镯赚到多少钱。
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  10. Write a function that shows how much money Rochelle has after selling $\begin{array}{r}b\end{array}$ bracelets, minus her investment in the kit.
      ::写一个函数 显示Rochelle卖了B手镯后 有多少钱 减去她对包子的投资
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  11. How many bracelets does Rochelle need to make before she breaks even?
      ::Rochelle要做多少手镯 才能破门而入?
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  12. If she buys a $50 display case for her bracelets, how many bracelets does she now need to sell to break even?
      ::如果她买了50美元的手镯展示盒 她现在要卖多少手镯才能扯平?

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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 。