2.3 按小数顺序确定的函数-interactive

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  小数定义函数

  Learning Objectives

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  • Understand the characteristics and structure of piecewise functions.
    ::理解片段函数的特性和结构 。
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  • Graph a piecewise function given the equation .
    ::给定的方程式, 图形化函数 。
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  • Interpret statements that use within a context over a specified interval.
    ::解释在特定间隔内使用上下文的语句。
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  • Write a piecewise function within a context.
    ::在上下文中写入一个片段函数 。
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  • Find the maximum value of a given interval across a piecewise function.
    ::在一个按片段函数中查找给定间隔的最大值。
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  • Understand the characteristics and structure of step functions.
    ::了解职级功能的特点和结构。
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  • Graph a step function given the equation.
    ::图形化一个阶梯函数,以给定方程式。

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  Introduction: Leasing a Car
  ::导言:汽车租赁

  

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  The payment plan for leasing a car is a piecewise function.
  ::租车的付款计划是一种零碎的功能。

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  Mathematicians, engineers, scientists, and many more professionals model the real world using functions. However, the world is a complicated place, and many relationships cannot be modeled by one function alone. These scenarios are where piecewise functions come into play. In this lesson , piecewise functions model scenarios in business, science, and more.
  ::数学家、工程师、科学家和更多的专业人士用功能模拟真实世界。 然而,世界是一个复杂的地方,许多关系不能单靠一个函数来模拟。 这些情景是小数函数发挥作用的地方。 在这个教训中,小数函数模型在商业、科学等领域的情景。

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  A simple example of the need for piecewise functions occurs when you lease a car.  To lease a car  is to essentially rent the car for a period of time.
  ::租车时,需要计件功能的一个简单例子就是租车。租车基本上是租车一段时间。

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  Example
  ::示例示例示例示例

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  You lease a car for four years for  $\begin{align*}$8,450\end{align*}$  with a mile limit of $\begin{align*}10,000\end{align*}$  miles. This limit means that if you drive more than $\begin{align*}10,000\end{align*}$  miles, you will have to pay a fee. For every mile you drive over $\begin{align*}10,000\end{align*}$  miles, you will have to pay  $\begin{align*}$0.30\end{align*}$  per mile.
  ::租车四年,租车8,450美元,限程10,000英里。这意味着,如果开车超过10,000英里,您必须支付费用。每开一万英里,您必须支付每英里0.30美元。

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  How could you write a function that will determine how much you will pay based on the number of miles driven? Is one function enough, or would you need more than one to model all possible outcomes?
  ::您如何写出一个函数来决定您要支付多少根据行驶英里数计算的金额 ? 一个函数是否足够, 或者您需要不止一个函数来模拟所有可能的结果 ?

   

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  Activity 1: Understanding Piecewise Functions
  ::活动1:了解零零散函数

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  Piecewise functions are used most often in programming where they are referred to as if-else statements or switch statements. A piecewise function is a function made up of two or more functions. The function needed is often determined by the context of the scenario. U se the following piecewise function to model the amount paid on a lease as a function of $\begin{align*}x\end{align*}$   miles driven over the lease.
  ::小数函数通常在编程中使用,当它们被称为 if-else 语句或切换语句时。小数函数是指由两个或两个以上函数组成的函数。需要的函数通常根据情景来决定。使用以下小数函数来模拟租赁所付金额,以x英里驱动租赁。

  

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  The conditions act like an air traffic controller sending inputs to the correct rule. T he input/output pair of a piecewise function is expressed the same  way that they are expressed for  any function:
  ::条件表现为空中交通控制器向正确规则发送输入。 片段函数的输入/输出对表示方式与任何函数的表达方式相同:

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  $$\begin{align*}f(7,000) = 8,450\end{align*}$$
  ::f( 7 000) = 8 450

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  In a pure math context, this means that if the input is $\begin{align*}7,000\end{align*}$  (which is between   and $\begin{align*}10,000),\end{align*}$  that input will be sent to the associated function rule . In this case, the function rule associated with the input $\begin{align*}7,000\end{align*}$  is  $\begin{align*}f(x) = 8,450.\end{align*}$  This rule means that  the output will always be $\begin{align*}8,450.\end{align*}$  In the context of the situation, if  you drive $\begin{align*}7,000\end{align*}$  miles,  you will have to pay  $\begin{align*}$8,450.\end{align*}$  If  you drive any number of miles $\begin{align*}10,000\end{align*}$  miles or less , you will have to pay  $\begin{align*}$8,450.\end{align*}$
  ::在纯数学背景下,这意味着如果输入为7 000(介于10,000至10,000之间),输入将被发送到相关函数规则中。 在这种情况下,输入 7 000 的函数规则是 f(x) = 8 450 。 这意味着输出始终为 8 450 。 在这种情况下,如果您驾驶7 000 英里,您必须支付 8 450 美元。如果您驾驶的英里为 10 000 英里或更小,您必须支付 8 450 美元。

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  $$\begin{align*}f(13,000) = 9,350\end{align*}$$
  ::f(13 000)=9 350

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  If  the input is $\begin{align*}13,000\end{align*}$  (which is greater than $\begin{align*}10,000),\end{align*}$  that input will be sent to the associated function rule. In this case, the function rule associated with the input $\begin{align*}13,000\end{align*}$  is  $\begin{align*}f(x) = 8,450 + 0.30(x - 10,000).\end{align*}$
  ::如果输入为 13 000( 大于 10,000) , 则该输入将被发送到相关函数规则 。 在此情况下, 与输入13 000 相关的函数规则是 f( x) = 8, 450+0. 30( x- 10,000) 。

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  $$\begin{align*}f(\boldsymbol{13,000}) = 8,450 + 0.30(\boldsymbol{13,000} - 10,000) = 9,350\end{align*}$$
  ::f(13 000)=8 450+0.30(13 000-10 000)=9 350

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  If  you drive $\begin{align*}13,000\end{align*}$  miles,  you will have to pay  $\begin{align*}$9,350.\end{align*}$   You will have to pay the  $\begin{align*}$8,450\end{align*}$  for the lease and the fee of  $\begin{align*}$0.30\end{align*}$  per mile over the $\begin{align*}10,000-\end{align*}$ mile limit. Since  you drove $\begin{align*}13,000\end{align*}$  miles,  you drove  $\begin{align*}13,000 - 10,000 = 3,000\end{align*}$  miles over the limit.  The fee is  $\begin{align*}$3,000\times 0.3 = $900.\end{align*}$
  ::如果你驾驶13 000英里,你必须支付9 350美元。你必须支付租金8 450美元和超过10 000英里界限的每英里0.30美元的费用。由于你驾驶13 000英里,你驾驶13 000-10 000英里=3 000英里,费用是3 000x0.3=900美元。

  iecewise-function-models" quiz-url="https://www.ck12.org/assessment/ui/embed.html?test/view/5eecd25fcb07a06c1b239a1f&collectionHandle=algebra&collectionCreatorID=3&conceptCollectionHandle=algebra-:iecewise-function-models&mode=lite" test-id="5eecd25fcb07a06c1b239a1f">

   

   

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  Activity 2 : Graphing a Piecewise Function
  ::活动2:绘制一小片函数图

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  Piecewise functions represent complicated information, and graphs can help make sense of it. Consider how to graph the example from the introduction. There are many ways to graph a piecewise function, but  one way to  begin is by creating a table of values to graph each part of the function individually.
  ::小数函数代表复杂的信息, 图表可以帮助理解它。 考虑如何从引言中绘制示例。 有多种方法可以绘制一个小数函数, 但开始的方法之一是创建一个数值表, 逐个绘制函数的每个部分 。

  

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  The graph above represents the constant function  $\begin{align*}f(x) = 8,450\end{align*}$   over the domain  $\begin{align*}[0,10,000].\end{align*}$  The solid circle represents the endpoints of the domain .
  ::上图表示域的常数 f(x) = 8,450。固态圆表示域的终点。

  

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  In the graph of  $\begin{align*}f(x) = 8,450 + 0.30(x - 10,000)\end{align*}$  over the domain  $\begin{align*}x > 10,000,\end{align*}$   there is an open circle at the point  $\begin{align*}(10,000, 8,450).\end{align*}$  The value at $\begin{align*}10,000\end{align*}$  is sent to the rule  $\begin{align*} f(x) = 8,450.\end{align*}$  However,  there is an open circle because the values between $\begin{align*}10,000\end{align*}$  and $\begin{align*}10,001\end{align*}$  are included in the graph. T he following graph combines both parts of the function:
  ::F(x) = 8,450+0.30(x- 10,000) 上方域 x > 10,000, 点上有一个开放圆(10,000, 8, 450) , 值为10,000 的值被发送到规则 f(x) = 8, 450。 但是, 则有一个开放圆, 因为图中包含10,000 和 10, 001 之间的值。 下图将函数的两个部分组合在一起 :

  

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  Example
  ::示例示例示例示例

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  A shipping company charges based on the weight of the package in pounds. The following function determines the cost of shipping a package,  $\begin{align*}C(x):\end{align*}$
  ::航运公司根据包件重量(磅)收取费用。

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  $\begin{align*}C(x) = \begin{cases} 4.50, & 0<x\le 3 \\ 0.5(x-3)+4.50, & 3<x\le10 \\ 0.5{(x-10)}^{2}+7.50, & 10<x\le12 \\ \end{cases}\end{align*}$
  ::C(x)4.50,0 <x%30.5(x-3)+4.50,3 <x%100.5(x-10)2+750,10 <x____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  Use the interactive below to graph the function by ploting points or using knowledge of function transformations .
  ::使用下方的交互功能,通过绘图点或使用函数转换知识绘制函数图。

  iecewise-function-models" quiz-url="https://www.ck12.org/assessment/ui/embed.html?test/view/5eecd933b0d9dff3131cfcae&collectionHandle=algebra&collectionCreatorID=3&conceptCollectionHandle=algebra-:iecewise-function-models&mode=lite" test-id="5eecd933b0d9dff3131cfcae">

   

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  Activity 3 : Writing a Piecewise Function
  ::活动 3: 写小写函数

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  One of the most common uses of piecewise functions comes from business and economics in the form of tax brackets . In the United States, taxes are paid to the U.S. government by citizens 18 years of age and older. Taxes pay for government programs like Medicare, public education, infrastructure, and much more.
  ::单件功能最常用的用途之一是以税项括号形式出现的商业和经济学。 在美国,18岁及以上公民向美国政府纳税。 政府计划(如Medicare 、 公共教育、基础设施等等)的税收支付更多。

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  The tax brackets below  were  used to determine how much an individual living by themselves would pay in taxes in 2019. Where taxes get tricky is that each bracket below only applies to the amount of money made that falls within that bracket: 
  ::以下的税额括号用于确定2019年一个人独自生活将支付多少税款。 当税收变得棘手时,下面的每个税额括号只适用于属于这一括号内的货币数额:

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  • If you  made $9,000 in 2019, you would only  have  to pay $900 in taxes, which is 10% of the $9,000.
    ::如果你在2019年赚到9000美元, 你只需付900美元的税, 也就是9000美元的10%。
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  • If you  made $12,000, you would have   to pay $1,249.50 in taxes. You would pay 10% on the first $9,525 and then 12% on only the money you made above $9,525.
    ::如果你赚了12 000美元,你就得缴纳1 249.50美元的税款。你会支付前9 525美元的10%,然后只支付9 525美元以上的12%。

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  Additionally, assume that the next tax bracket does not start until one cent over the maximum value. An income of $13,600 will be subject to the 10% tax rate , but an income of $13,600.01 will be subject to the 12% tax rate.
  ::此外,假设下一个税级不会开始,直到最高值的1%。 13,600美元的收入将按10%的税率计算,但13,600.01美元的收入将按12%的税率计算。

  Tax Rate	Head of Household
  10%	$0 to $13,600
  12%	$13,600 to $51,800
  22%	$51,800 to $82,500
  24%	$82,500 to $157,500
  32%	$157,500 to $200,000
  35%	$200,000 to $500,000
  37%	$500,000 or more

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  Example
  ::示例示例示例示例

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  Convert the federal tax bracket above to a piecewise function.
  ::将上面的联邦税级转换成一个计件函数。

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  Since there are seven total tax brackets, there will be seven rules in  this piecewise function. Each rule will also need to take into account the lower brackets. The first bracket is pretty simple, for all incomes from $0 up to $13,600,  a taxpayer  would have to pay  10% of the income.
  ::由于共有7个税括号,本项功能中将有7条规则。 每条规则也需要考虑低括号。 第一个括号很简单,对于从0美元到13 600美元的所有收入,纳税人必须支付收入的10%。

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  $$\begin{align*}f(x) = 0.10x, \:\:\:0 ≤ x ≤ 13,600\end{align*}$$
  ::f(x)=0.10x,0xx/13 600

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  The second bracket is where it gets tricky.  A taxpayer would pay 10% on the first $13,600 and then 12% on only the amount of money  earned above $13,600.
  ::第二个括号是困难所在。 纳税人将支付前13,600美元的10%,而后仅支付13,600美元以上收入的12%。

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  $$\begin{align*}f(x) = 0.10(13,600) + 0.12(x - 13,600)\end{align*}$$
  :xx)=0.10(13 600)+0.12(x-13 600)

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  13,600  is in the function because it would not be possible for a taxpayer to be in this tax bracket if  they  did not make at least $13,600. This expression simplifies to the following:
  ::13 600名纳税人正在履行这一职能,因为如果纳税人没有至少挣到13 600美元,则不可能进入这一税项。

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  $$\begin{align*}f(x) = 1360 + 0.12(x - 13,600), \:\:\:13,600 < x ≤ 51,800\end{align*}$$
  ::f(x) = 13660+0.12(x-13 600),13 600 < x%51 800

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  In the third bracket,  a taxpayer  would pay 10% of the first $13,600  they  earned, 12% on any money that  they earned over $13,600 up to $51,800, and 22% on any money  they   earned over $51,800.
  ::在第三档,纳税人将支付头13 600美元收入的10%,其收入超过13 600美元的任何款项的12%,最高达51 800美元,其收入超过51 800美元的任何款项的22%。

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  $$\begin{align*}f(x) = 0.10(13,600) + 0.12(51,800 - 13,600) + 0.22(x - 51,800)\end{align*}$$
  ::f(x)=0.10(13 600)+0.12(51 800-13 600)+0.22(x-51 800)

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  This expression simplifies to the following:
  ::此表达式简化为:

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  $$\begin{align*}f(x) = 5,944 + 0.22(x - 51,800), \:\:\:51,800 < x ≤ 82,500\end{align*}$$
  ::f(x) = 5 944+0.22(x-51 800)、51 800 <x182 500

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  Answer the questions  below to finish this example and practice writing piecewise functions in other scenarios. 
  ::回答以下问题,以完成这一例子,并练习在其他情景中逐个撰写函数。

  iecewise-function-models" quiz-url="https://www.ck12.org/assessment/ui/embed.html?test/view/5eece2e11c68678137c27524&collectionHandle=algebra&collectionCreatorID=3&conceptCollectionHandle=algebra-:iecewise-function-models&mode=lite" test-id="5eece2e11c68678137c27524">

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  Discussion Question : Why aren’t the domain and range of the tax bracket function all real numbers?
  ::讨论问题:为什么税级的域和范围不能发挥所有实际数字的作用?

   

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  Activity 4 : Step Functions
  ::活动4:职级职能

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  A parking garage lists the following rates:
  ::停车场列出以下费率:

  Time	Total Fee
  0 to 1 hour	$6
  1 to 3 hours	$15
  3 to 5 hours	$22
  5 to 7 hours	$28
  7+ hours	$35

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  T his can be written as a piecewise function using the same strategy used in the previous activity. However, keep in mind that the prices do not accumulate. If you park for  $\begin{align*}5\tfrac{1}{2}\end{align*}$  hours, you pay $28. Additionally, assume that, aside from 0, the first number in each  part of the  domain is inclusive, and the second number is exclusive. You would pay $15 if you parked for exactly 60 minutes.
  ::此功能可以使用上一个活动中使用的相同策略来写成一个片段函数。 但是, 请记住, 价格不会累积 。 如果您在512 小时内停车, 您支付 28 美元 。 此外, 假设除 0 以外, 域内每个部分的第一个数字是包含的, 第二个数字是独家的。 如果您在60 分钟内停车, 您将支付 15 美元 。

  

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  The g raph of this piecewise function illustrates the reason why it is called a step function.
  ::此片段函数的图示显示了为什么它被称为步脚函数的原因 。

  

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  A step function is a piecewise function that is made up exclusively of constant functions. The parent step function uses the notation $\begin{align*}f(x) = [[x]].\end{align*}$  This function means that any input value is rounded down to the nearest integer .
  ::职档函数是指完全由常数函数组成的片段函数。父级阶梯函数使用 f(x)=[[[x]]。此函数意味着任何输入值都被四舍五入到最接近的整数。

  

  iecewise-function-models" quiz-url="https://www.ck12.org/assessment/ui/embed.html?test/view/5eed1e3da4bf50c411188b09&collectionHandle=algebra&collectionCreatorID=3&conceptCollectionHandle=algebra-:iecewise-function-models&mode=lite" test-id="5eed1e3da4bf50c411188b09">

    

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  Extension: Floor and Ceiling Functions
  ::延长:下限和上限职能

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  The parent step function is commonly used in programming and known as the floor function. The floor function uses the notation  $\begin{align*}f(x) = \lfloor x \rfloor.\end{align*}$  In addition to the floor function, the ceiling function is similar and commonly used step function where the input value is rounded up to the nearest integer. The notation for the ceiling function is  $\begin{align*}f(x) = \lceil x \rceil.\end{align*}$
  ::父级阶梯函数在编程中通常使用,称为下层函数。下层函数使用编号 f(x) 。除了下层函数外,上限函数与输入值四舍五入到最接近的整数的上限函数相似,而且常用。上限函数的标记是 f(x) 。

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  Discussion Questions :
  ::讨论问题:

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  • What would the difference be between  $\begin{align*}\lfloor 2.3 \rfloor\end{align*}$  and  $\begin{align*}\lceil 2.3 \rceil?\end{align*}$  
    ::2.3和2.3之间有什么区别?
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  • What would the difference be between  $\begin{align*}\lfloor -5 \rfloor\end{align*}$  and $\begin{align*}\lceil -5 \rceil?\end{align*}$  
    ::5和5有什么区别?
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  • Can you think of any real-world situations where you would want to round up or round down a given number?
    ::你能想到任何现实世界的情况吗? 在那里你想整理或整理一个特定的数字?

   

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  Wrap-Up: Review Questions
  ::总结:审查问题

  iecewise-function-models" quiz-url="https://www.ck12.org/assessment/ui/embed.html?test/view/5eed2541659483e5f6902467&collectionHandle=algebra&collectionCreatorID=3&conceptCollectionHandle=algebra-:iecewise-function-models&mode=lite" test-id="5eed2541659483e5f6902467">

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     Summary
  ::摘要

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  • A piecewise function is a function that is made up of 2 or more functions defined on different intervals.
    ::片段函数是指由按不同间隔界定的2个或2个以上函数组成的函数。
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  • A step function is a type of piecewise function where each piece/step is a horizontal line.
    ::职档函数是一种片段函数类型,其中每个片段/步骤为水平线条。