节: 函数的域域和范围 | 1.2 职能的域域和范围

章节大纲

You and your 3 friends are debating which theater to choose for a trip to the movies on Friday. There are 4 theaters in town, all with different prices for tickets. The most expensive theater charges $12.50 per person, but has stadium seating and a screen 5 stories tall. The cheapest is only $5.50 per ticket, but only plays older movies and is much less comfortable. The other two are between the extremes at $8.50 and $9.75 each.
::你和你的三个朋友正在辩论周五哪个剧院可以选择去看电影。城里有四个剧院,所有剧院的票价都不同。最昂贵的剧院每人收费12.50美元,但体育场座位和屏幕5层高。最便宜的是每张罚单只有5.50美元,但只播放老电影,更不舒适。另外两个在最高区之间,每张价格为8.50美元和9.75美元。

If you were to graph the total costs for the 4 of you based on which theater you choose, which values would represent the domain and which the range ?
::如果您要根据您选择的剧院、哪个值代表域和哪个区域来计算你们4人的总成本?

Domain and Range
::域和范围

The set of all possible input values for the independent variable of a function is called the domain . The domain can be expressed: in words, as a set, or as an inequality .
::函数独立变量的所有可能输入值组称为域。域可以表示: 文字、集或不平等。

The range of a function consists of all the values of the dependent variable resulting from all the values of the independent variable. The range can be expressed: in words, as a set, or as an inequality.
::函数的范围由独立变量所有值产生的附属变量的所有值组成。该范围可以表示:用文字表示、作为一组表示、或作为不平等表示。

Consider the function represented by the ordered pairs $\begin{aligned} (x, y) : {(- 3, 1), (0, 4), (3, 8), (6, 13), (9, 19)} \end{aligned}$ .
::考虑定购对子(x,y):{(-3,1,(0,4),(3,8),(6,13),(9,19)的功能。

What is the domain of the function and what is the range?
::函数的域是什么? 范围是什么?

The function maps each value of $\begin{aligned} x \end{aligned}$ , the independent variable, to a value of $\begin{aligned} y \end{aligned}$ , the dependent variable.
::函数绘制 x 的每个值、独立的变量、 y 的值、依附变量。

The domain consists of the group of values (-3, 0, 3, 6, 9). Normally these are arranged from least to greatest value.
::域由一组值( 3, 0, 3, 6, 9) 组成, 通常这些值由最小值排列为最大值。

The range consists of the group of values (1, 4, 8, 13, 19). Normally these are arranged from least to greatest value.
::范围由一组值组成(1、4、8、13、19),通常这些值从最低值排列为最高值。

Note that both the were represented as a set of values.
::请注意,这两个数值都是一组数值。

Take the following functions:
::承担以下职能:

$\begin{aligned} y = x^{2} \end{aligned}$
::y=x2 y=x2

$\begin{aligned} y = \sqrt{x} \end{aligned}$
::y=x y=x

The domain of (a) is the set of all real numbers. The range of the function is the set of all real numbers greater than or equal to zero. This is the case because every $\begin{aligned} y \end{aligned}$ value is the square of an $\begin{aligned} x \end{aligned}$ value, which is always positive. The range could be written as $\begin{aligned} y \geq 0 \end{aligned}$ .
:a) 的域是所有实际数字的一组。函数的范围是所有实际数字大于或等于零的一组。这是因为每个y值都是 x 值的正方形,而 x 值总是正数。范围可以写为 y_0 。

For function (b), because the function takes the square root of every $\begin{aligned} x \end{aligned}$ , values of $\begin{aligned} x \end{aligned}$ must be restricted to values greater than or equal to 0. This means the domain is the set of all non-negative real numbers. Another way to indicate this is to say the domain is all $\begin{aligned} x \geq 0 \end{aligned}$ . The range is therefore all non-negative real numbers, i.e., the range is all $\begin{aligned} y \geq 0 \end{aligned}$ .
::对于函数 (b) , 因为函数取每 x 的平方根, x 的值必须限于大于或等于 0 的值。这意味着域是所有非负真实数字的一组。另一个表示方式是表示域是全部 x=0。因此, 范围是所有非负真实数字, 即范围是全部 y=0 。

Note that the domain and ranges were described using inequalities.
::请注意,在说明领域和范围时使用了不平等。

Many times, the domain or range of a function is all numbers except for a particular number. Take the function $\begin{aligned} y = \frac{2}{x} \end{aligned}$ .
::许多次,函数的域或区域是除特定数字以外的所有数字。请使用函数 y=2x。

For this function, we can choose any $\begin{aligned} x \end{aligned}$ value except $\begin{aligned} x = 0 \end{aligned}$ , since we cannot divide a number by zero. Therefore the domain of the function is the set of all real numbers except $\begin{aligned} x = 0 \end{aligned}$ , or the domain is $\begin{aligned} x \neq 0 \end{aligned}$ .
::对于此函数,我们可以选择除 x=0 以外的任何 x 值,因为我们不能将数字除以零。因此,该函数的域是除 x=0 或域是 x0 外所有实际数字的集合。

The range is also restricted to the non-zero real numbers because the fraction can never equal zero. The range is $\begin{aligned} y \neq 0 \end{aligned}$ .
::该范围还限于非零实际数字,因为分数永远不能等于零。范围为y0。

Examples
::实例

Example 1
::例1

Earlier, you were asked about the values of domain and range of the cost for seeing a movie when comparing 4 different movie theaters.
::早些时候,有人问过你在比较4个不同的电影院时, 看电影的域值和成本范围。

One function that describes this situation would be: $\begin{aligned} P (t) = 4 t \end{aligned}$ , where $\begin{aligned} P (t) \end{aligned}$ would be total Price based on per-ticket price, and it is equal to $\begin{aligned} 4 \times ticket price \end{aligned}$ .
::描述这种情况的一个功能是:P(t)=4t,其中P(t)将按每张票价计算总价格,等于4x票价。

In this case, your domain would be: {$5.50, $8.50, $9.75, $12.50} since these are the prices you would input in place of the independent variable, $\begin{aligned} t \end{aligned}$ to get the total price for each theater. The range would be: {$22.00, $34.00, $39.00, $50.00} since these are the output values given by the dependent variable, $\begin{aligned} y \end{aligned}$ .
::在此情况下,您的域名将是 {5.50, 8.50, 9.75, 12.50},因为这些是您将投入的价格代替独立的变量, t 以获得每个剧院的总价格。范围是 {22.00, 34.00, 39.00, $50.00}, 因为这些是依附变量y 给出的产出值。

Example 2
::例2

Determine if each relation is a function:
::确定每一关系是否为函数:

(-1, 4), (0, 3), (1, 5), (1, 7), (2, 15)

Not a function, since there are two different $\begin{aligned} y \end{aligned}$ values for the same $\begin{aligned} x \end{aligned}$ value (1).
::函数不是函数,因为相同 x 值 (1) 有两个不同的y 值。

:1,4,0,3),(1,5),(1,7),(2,15)不是函数,因为相同x值有两个不同的y值(1)。

$\begin{aligned} y = x \end{aligned}$

Is a function. Since $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ are the same number (equal), any number chosen for $\begin{aligned} x \end{aligned}$ will have one and only one $\begin{aligned} y \end{aligned}$ .
::函数。由于 x 和 y 是相同的数字(相等),因此为 x 选择的任何数字将有一个,只有一个y。

::y=x 是一个函数。由于 x 和 y 是相同数字(相等),因此为 x 选择的任何数字将有一个,只有一个y。

Example 3
::例3

State the domain and range of each relation in Example 2.
::在例2中,说明每一关系的范围和范围。

(-1, 4), (0, 3), (1, 5), (1, 7), (2, 15)

The domain is the set of $\begin{aligned} x \end{aligned}$ values: {-1, 0, 1, 2}. The range is the set of $\begin{aligned} y \end{aligned}$ values: {3, 4, 5, 7, 15}
::域是 x 值组 : {-1, 0, 1, 2}。范围是 y 值组 : {3、4、 5、 7、 15}

:-1, 4, 0, 3, 1, 5, 1, 5, 7, 2(, 7), (2, 15) 域是 x 值组 : {-1, 0, 1, 2} 。范围是 y 值组 : 3, 4, 5, 7, 15}

$\begin{aligned} y = x \end{aligned}$

The domain or $\begin{aligned} x \end{aligned}$ value can be any real number . $\begin{aligned} D : R \end{aligned}$ . The range or $\begin{aligned} y \end{aligned}$ value can therefore also be any real number. $\begin{aligned} R : R \end{aligned}$
::域或 x 值可以是任何实际数字。 D: R。因此, 范围或 y 值也可以是任何实际数字。 R: R: R

::y=x 域或 x 值可以是任何实际数字。 D: R。因此,范围或 y 值也可以是任何实际数字。 R: R: R:

Example 4
::例4

Imagine a playground full of children where all of the boys are wearing baseball caps. The function “boys in the playground” could be used to identify kids wearing hats. In this case, which group would be the domain, and which the range? In other words, what group of kids is the input, and what group the output?
::想象一下一个充满儿童、所有男孩都戴着棒球帽的操场。 “操场男孩”的功能可以用来识别戴帽子的孩子。在这种情况下,哪个群体是球场,哪个区域是球场? 换句话说,输入的是哪个儿童群体,产出是哪个群体?

In this example, “boys in the playground” is the domain, since you would use that group to identify hat-wearers.
::在这个例子中,“操场男孩”是域名,因为你会利用该组来识别戴帽子的人。

Example 5
::例5

Would the example from Example 4 be a function if used backwards? Which group would be the domain and which the range in that case?
::例4中的示例如果倒着使用,是否是一个函数?如果倒着使用,哪个组是域,在哪个组是域? 在这种情况下,哪个区域是范围?

It could be, it depends on whether or not there are girls wearing hats! If there are girls in the playground wearing hats, then inputing “kids wearing hats” would not return only “boys in the playground”. In either case, however, “kids wearing hats” would be in domain in this version of the function.
::这可能取决于是否有女孩戴帽子!如果在操场有女孩戴帽子,那么输入“孩子戴帽子”不会只返回“在操场的男孩”,但是,无论在哪种情况下,“孩子戴帽子”都属于这一职能的范畴。

Review
::回顾

For #1-2, determine the domain and the range of the relations.
::对于 #1-2, 确定关系域和范围。

Relation: {(0, 4) (3, 20) (90, 33)}
::关系:{(0, 4) (3, 20) (90, 33)}

Relation: {(3, -4) (6, 37) (10, -10) (-31, 2)}
::关系:{(3, 3, 4) (6, 37) (10, 10) (31, 2) }

Tina’s car travels about 30 miles on one gallon of gas. She has between 10 and 12 gallons of gas in the tank. Find the domain and range of the function to calculate how far she can drive.
::蒂娜的汽车在一加仑的气体上行驶约30英里。她的油箱里有10到12加仑的气体。找到功能的域域和范围来计算她能开多远。

Joe and three of his friends plan on going bowling and plan on bowling one or two games each. Each game costs $2.75. Find the domain and range of the function calculating the cost of the trip.
::乔和他的三个朋友计划去打保龄球并计划每人打一两场保龄球每场比赛要花2.75美元找出计算行程成本的功能的域域和范围

Bob had a summer job that paid $10.00 per hour and he worked between 20-25 hours every week. His weekly salary can be modeled by the equation $\begin{aligned} S = 10 h \end{aligned}$ , where $\begin{aligned} S \end{aligned}$ is his weekly salary and $\begin{aligned} h \end{aligned}$ is the number of hours he worked per week. What is the independent variable for this problem? Describe the domain and range for this problem.
::Bob的暑期工作每小时10.00美元,每周工作20-25小时。他的每周工资可以S=10小时的等式制成,S=10小时是每周工资,h是每周工作小时数。这个问题的独立变量是什么?描述这个问题的域和范围。

What does each value in the ordered pair (20, 200) mean in context of #5?
::订货配对(20,200)中的每种值在5号上下文中意味着什么?

Which group of students represents the domain, and which the range, in these ordered pairs? (Jim, Kitty) (Joe, Betty) (Brian, Alice) (Jesus, Anissa) (Ken, Kelli)
::这些定购的对子中,哪一组学生代表域名和范围? (Jim, Kitty) (Joe, Betty) (Brian, Alice) (耶稣,Anissa) (Ken, Kelli) (Ken, Kelli) (Jim, Kitty) (Joe, Betty) (Brian, Alice) (耶稣, Anissa) (Ken, Kelli)

For #8-15, state the domain and range.
::在8-15中, 说明域域和范围。

$\begin{aligned} y = | x | \end{aligned}$
::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

$\begin{aligned} x = | y | \end{aligned}$
::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

$\begin{aligned} y = x^{2} \end{aligned}$
::y=x2 y=x2

$\begin{aligned} x = y^{2} \end{aligned}$
::x=y2x=y2

Review (Answers)
::回顾(答复)

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。

章节大纲

Domain and Range ::域和范围

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Domain and Range
::域和范围

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)