Course: 1.16 关系和职能 | The Way To Learn

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关系和职能

Introduction
::导言

Suppose you want to predict how much it would cost to see a movie at the theater. You text a number of friends who've recently been to the movies, and ask how much it cost them. Here are their responses:
::假设你想预测在剧院看电影要花多少钱。你发短信给最近看过电影的一些朋友, 并问他们花了多少钱。以下是他们的答复:

"$14.50 "

"$8.75 + $3.50 for popcorn"
::"8.75美元+3.50美元的爆米花"

"five bucks - dollar theater"
::"五块钱 -一块钱剧院"

"$17.50 :'-( broke now"
::"17.50美元:" -(现在破了)"

"$12.75 loved the 3D!"
::"12.75美元爱3D!"

Can you accurately predict the cost of going to a movie from these responses? Why or why not?
::你能从这些回应中准确预测去看电影的成本吗?为什么或为什么?

Relations and Functions
::关系和职能

Consider two situations shown in the boxes below:
::考虑以下方框所示两种情况:

Situation 1: You are selling candy bars for a school fundraiser. Each candy bar costs $3.	Situation 2: You collect data from several students in your class about their ages and heights, and get the following information: (18,65"), (17,64"), (18,67"), (18,68"), (17,66").

In the 1st situation, let the variable $\begin{aligned} x \end{aligned}$ represent the number of candy bars you sell, and let $\begin{aligned} y \end{aligned}$ represent the amount of money you make. If you sell $\begin{aligned} x \end{aligned}$ candy bars, you'll make $\begin{aligned} y = 3 x \end{aligned}$ dollars. For example, if you sell 25 candy bars, you'll make 3(25) = $75. Notice you can use the number of candy bars you sell to predict how much money you'll make.
::在第一种情况下, 让变量 x 代表您出售的糖果棒数量, 让您代表您挣的钱数量。如果您卖了糖果棒, 您就会赚到 y= 3x 美元。例如, 如果您卖了 25 个糖果棒, 您就会赚到 3( 25 ) = 75 美元。请注意, 您可以使用您卖的糖果棒数量来预测您能赚到多少钱。

Now consider the 2nd situation. Can you similarly use the data you collected to predict specific height, based on age?
::现在考虑第2种情况。您能否同样使用您收集的数据来预测特定高度, 以年龄为基础 ?

No, you cannot make such a prediction in this case. For example, if a student is 18 years old, there are a number of possible heights that student could be.
::不,您无法在这种情况下做出这样的预测。例如,如果学生年满18岁,学生可能身高可能很多。

The 1st situation is an example of a function , and the 2nd example is not a function.
::第一种情况是函数的一个例子,第二种情况不是函数。

Definition of a Function

::职能的定义

A function is a relationship in which each input number corresponds to one and only one output number.

In the 1st situation, for each different number of candy bar sales you input, there is one and only one output number representing your profit.
::在第一种情况下,对于每个不同的糖果棒销售量你输入, 有一个,只有一个产出数代表你的利润。

In the 2nd situation, if you input "18 years," there are multiple outputs, so you can't identify a specific relationship between age and height.
::在第二种情况下,如果你输入“18岁”, 就会有多重输出, 所以您无法确定年龄和身高之间的特定关系。

It is important to note that both situations above are relations. A relation is a pairwise relationship between two sets of numbers or data. For example, in the 2nd situation, we created a relationship between students’ ages and heights by writing each student’s information as an ordered pair . In the 1st situation, there is a relationship between the number of candy bars you sell and the amount of money you make. The 1st example is different from the 2nd because it represents a function : every $\begin{aligned} x \end{aligned}$ is paired with only one $\begin{aligned} y \end{aligned}$ .
::必须指出,上述两种情况都是关系。一种关系是两组数字或数据之间的一种对称关系。例如,在第二种情况下,我们通过将每个学生的信息写成一对定做的,在学生的年龄和身高之间建立了一种关系。在第二种情况下,你卖的糖果条数与你赚的钱额之间存在一种关系。第一个例子与第二个不同,因为它代表了一个函数:每个x只配一个y。

Functions may be represented in many ways. Some of the most common ways to represent functions include sets of ordered pairs, equations, and graphs. The table below in Example 1 shows the same function depicted in three different ways.
::函数可以以多种方式表示。一些最常见的函数表达方式包括一对一对、一对一对、一方程式和图表。下面的表1显示了以三种不同方式描述的相同功能。

This video below provides an overview of how to determine if a relation is a function. It includes vocabulary definitions, the techniques of mapping and the vertical line test to determine if a relation is a function, and examples.
::以下这段视频概述了如何确定关系是否是一个函数,包括词汇定义、绘图技术和垂直线测试,以确定关系是否是一个函数,以及实例。

To further explore the idea of vertical line testing, see this PLIX: .
::为进一步探讨垂直线测试的设想,请参见PLIX:.

Examples
::实例

Example 1
::例1

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

Representation	Example
Set of ordered pairs	(1,3), (2,6), (3,9), (4,12) (a subset of the ordered pairs for this function)
Equation	$\begin{aligned} y = 3 x \end{aligned}$
Graph

Solution:
::解决方案 :

In the 1st representation above, we are given a set of ordered pairs. To verify that this is a function, we must ensure that each $\begin{aligned} x \end{aligned}$ -value is associated with a single $\begin{aligned} y \end{aligned}$ -value. In this example, the 1st number in each pair (the $\begin{aligned} x \end{aligned}$ -value) is different, so we can be certain that there are no cases in which a particular $\begin{aligned} x \end{aligned}$ is associated with more than one $\begin{aligned} y \end{aligned}$ .
::在以上第一个表达式中, 我们得到了一组有顺序的对子。为了验证这是一个函数, 我们必须确保每个 x 值与一个 Y 值相关。在此示例中, 每对的一号数( x 值) 不同, 因此我们可以确定没有特定 x 与一个 y 以上相关的情况。

In the 2nd representation, the equation of a line, it is apparent that any number put in place of $\begin{aligned} x \end{aligned}$ will result in a different $\begin{aligned} y \end{aligned}$ , since the $\begin{aligned} x \end{aligned}$ number is simply being multiplied by $\begin{aligned} 3 \end{aligned}$ .
::在第二个表示式,即一行的等式中,很明显,任何以 x 代替 x 的数值都会产生不同的y,因为x 数只是乘以3。

The 3rd representation above is a graph. A good way to determine whether a relation is a function when looking at a graph is by doing a "vertical line test." If a vertical line can be drawn anywhere on the graph such that the line crosses the relation in two places, then the relation is not a function. If all possible vertical lines cross the relation in only one place, then the relation is a function. This works because if a vertical line crosses a relation in more than one place, it means there must be two y values corresponding to one x value in that relation. For example, the graph above of $\begin{aligned} y = 3 x \end{aligned}$ shows it is a function because any vertical line that is drawn crosses the relation in only one place.
::上面的第3个表示式是一个图形。在查看一个图形时, 确定一个关系是否是一个函数的一个很好的方法就是进行“ 垂直线测试”。如果在图形的任何地方可以绘制一条垂直线, 这样一行横跨两个位置的关系, 那么关系就不是一个函数。如果所有可能的垂直线只跨越一个位置的关系, 那么关系就是一个函数。因为如果一条垂直线在不止一个位置上跨过一个关系, 它意味着该关系中必须有两个y值对应一个 x 值。例如, 上面的 y=3x 的图形显示它是一个函数, 因为所绘制的任何垂直线只跨越一个位置的关系。

Conversely, the graph below of $\begin{aligned} x \end{aligned}$ = $\begin{aligned} y \end{aligned}$ ² shows it is not a function because a vertical line can be drawn that crosses the relation in two places:
::反之, x = y2 下方的图表显示它不是一个函数,因为可以绘制垂直线,横跨两个位置的关系:

Example 2
::例2

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

a) (2, 4), (3, 9), (5, 11), (5, 12)	b) Relation defined as:

Solutions:
::解决办法:

a) (2, 4), (3, 9), (5, 11), (5, 12)
:a) (2,4),(3,9),(5,11),(5,12)

This relation is not a function because 5 is paired with 11 and with 12.
::这种关系不是一个功能,因为5与11和12对齐。

b) This relation is a function because every $\begin{aligned} x \end{aligned}$ is paired with only one $\begin{aligned} y \end{aligned}$ . A vertical line through the graph will always only encounter a single point.
:b) 这一关系是一个函数,因为每个x只配有一个y。图表中的一条垂直线总是只遇到一个点。

Example 3
::例3

Recall the problem in the Introduction about movie tickets. Does the data you received from your friends represent a function? Can you use the data to predict the cost of going to a movie?
::回顾开场白中关于电影票的问题。你从朋友那里得到的数据是否代表一个功能? 您能否用这些数据来预测去看电影的成本?

Solution:
::解决方案 :

If we were to organize the information we received into ordered pairs, it might look something like $\begin{aligned} (1, 14.5), \end{aligned}$ $\begin{aligned} (1, 8.75), \end{aligned}$ $\begin{aligned} (1, 5), \end{aligned}$ $\begin{aligned} (1, 17.5), \end{aligned}$ $\begin{aligned} (1, 12.75), \end{aligned}$ where each $\begin{aligned} x \end{aligned}$ -value represents the number of tickets bought, and each $\begin{aligned} y \end{aligned}$ -value represents the price.
::如果我们将我们收到的信息编成一对定购单,它可能看上去像(1,14.5)、1,8.75、1,5、1,17.5)、1,12.75,其中每个x值代表购买的机票数目,每个y值代表价格。

Since there are many different $\begin{aligned} y \end{aligned}$ -values for the only $\begin{aligned} x \end{aligned}$ -value, it is definitely not a function.
::由于唯一的 x 值存在许多不同的 Y 值,因此它绝对不是一个函数。

It should now be clear that the information received from friends' text messages cannot be used to accurately predict the cost of a movie.
::现在应该清楚的是,从朋友的短信中获得的信息不能用来准确预测电影的成本。

Example 4
::例4

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

a) $\begin{aligned} (- 1, 4), (0, 3), (1, 5), (1, 7), (2, 15) \end{aligned}$
:a) (-1,4),(0),(3),(1,5),(1,7),(2,15)

Solution:
::解决方案 :

There are two different "outputs" or $\begin{aligned} y \end{aligned}$ -values for the "input" or $\begin{aligned} x \end{aligned}$ -value of 1. Because we cannot know whether 1 should go with 5 or 7 at any given time, this relation is not a function.
::1的“投入”或X值有两个不同的“产出”或Y值。因为我们不能知道1在任何特定时间是5还是7, 这种关系不是函数。

b) $\begin{aligned} y = x \end{aligned}$
::b) y=x

Solution:
::解决方案 :

Since $\begin{aligned} y = x \end{aligned}$ , any time a number is chosen to represent $\begin{aligned} x \end{aligned}$ , that and only that number becomes $\begin{aligned} y . \end{aligned}$ From this it is apparent that each input has one and only one output, so this relation is a function.
::由于 y=x, 每当选择一个数字来代表 x, 而只有这个数字才会成为 y。从此可以明显看出, 每一个输入有一个输出, 只有一个输出, 因此此关系是一个函数。

c) $\begin{aligned} (2, 0), (4, - 1), (2.1, 4), (1, 4), (4, - 1) \end{aligned}$
:2,0,(4,-1),(2.1,4),(1,4),(4,-1)

Solution:
::解决方案 :

Don't be fooled! This is a function, as there is only one unique output for each input. The fact that both x- values 2.1 and 1 are associated with y value 4 does not mean that 2.1 and 1 don't have a specific associated value. Also, no matter how close two x 's (2 and 2.1, for instance) may be, if they are not exactly the same, they don't affect the definition of a function.
::不要被愚弄! 这是一个函数, 因为每个输入只有一个独特的输出。 x- 值 2. 1 和 1 都与 y 值 4 相关, 并不意味着 2. 1 和 1 没有特定的相关值。另外, 无论两个 x ( 2 和 2. 1 ) 的距离有多近, 如果它们不完全相同, 它们并不影响函数的定义。

d) $\begin{aligned} y = 4 x \end{aligned}$
::d) y=4x

Solution:
::解决方案 :

This is a function. Any value chosen for $\begin{aligned} x \end{aligned}$ has one and only one associated value for $\begin{aligned} y \end{aligned}$ (four times as big).
::这是一个函数。为 x 所选择的任何值为y(四倍大)只有一个且只有一个相关值。

e) $\begin{aligned} x = | y | \end{aligned}$
::e) xy

Solution:
::解决方案 :

This is not a function. This graph looks like a "<" with the vertex on the origin. Any positive value chosen for $\begin{aligned} x \end{aligned}$ will have two associated $\begin{aligned} y \end{aligned}$ - values. For instance, 4 = |-4| and 4 = |4|.
::这不是函数。此图表看起来像一个“ < ” , 上面有源头的顶点。为 x 所选择的任何正值都将有两个相关的 Y 值。例如, 4 = @ @ 4 @ 4} 和 4 = @ 4} 。

Summary
::摘要

A relation is a relationship between two sets of numbers or data.
::一种关系是两组数字或数据之间的关系。

A relation is also a function if every $\begin{aligned} x \end{aligned}$ is paired with only one $\begin{aligned} y \end{aligned}$ .
::如果每x只配上一个y,则关系也是一种函数。

Review
::回顾

What is the definition of a function?
::函数的定义是什么?

Can a function be written in the form $\begin{aligned} x = 3 y \end{aligned}$ instead of $\begin{aligned} y = 3 x \end{aligned}$ ?
::函数是否可以以 x=3y 而不是 y=3x 的形式写入 ?

Is it mandatory for a function to have both an input and an output?
::一项函数必须同时有输入和输出吗?

Can a statement be a function if there is only one input and output?
::如果只有一个输入和输出, 语句能否成为函数 ?

Give an example of a relation that is not a function, and explain why it is not a function.
::请举一个非函数关系的例子,并解释为什么它不是函数。

For Questions 6-14, identify each relation as either a function, or not a function:
::对于问题6-14,将每一关系确定为一项职能或非一项职能:

(2, 4) (4, 6) (6, 8) (3, 4) (5, 7) (8, 2)
(-1, 6) (0, 4) (-4, 0) (-1, -6) (-3, -8)

This relation with input values on the left and output on the right:

::与左侧输入值和右侧输出值的关系 :

This relation with input values on the left and output on the right:

::与左侧输入值和右侧输出值的关系 :

(Jim, Kitty) (Joe, Betty) (Brian, Alice) (Jesus, Anissa) (Ken, Kelli)
:吉姆,凯蒂) (乔,贝蒂) (布里安,爱丽丝) (耶稣,阿尼萨) (肯,凯利)

(Jim, Alice) (Joe, Alice) (Brian, Betty) (Jim, Kitty) (Ken, Anissa)
:吉姆,爱丽丝) (乔,爱丽丝) (布里安,贝蒂) (吉姆,凯蒂) (肯,阿尼萨)

At a prom dance, each boy pins a corsage on his date. Is this an example of a function?
::在舞会舞会上,每个男孩都在他的约会上插上皮带。这是功能的一个例子吗?

Later, at the same dance, Cory shows up with two dates. Does this change the answer?
::后来在同一场舞会上,Cory出现了两个日期。这改变了答案吗?

Review (Answers )
::回顾(答复)

Please see the Appendix.
::请参看附录。

Section outline

General

关系和职能

Introduction ::导言

Relations and Functions ::关系和职能

Definition of a Function ::职能的定义

Examples ::实例

Summary ::摘要

Review ::回顾

Review (Answers ) ::回顾(答复)

Introduction
::导言

Relations and Functions
::关系和职能

Definition of a Function

::职能的定义

Examples
::实例

Summary
::摘要

Review
::回顾

Review (Answers )
::回顾(答复)