Section: 职能职能职能职能职能职能职能职能职能职能职能职能职能职能职能 | 1.1.1 职能职能

Section outline

Suppose you wanted to get cash for your $250 stash of nickels, dimes and quarters that you had accumulated over many years. To see where you could get the best deal, you went online to check out what banks would give you for your coins, and you got the following redemption information:
::假设你想用你积累了多年的250美元的硬币、硬币、硬币和住宅获得现金。为了看哪里能得到最好的交易,你上网查看银行会给你的硬币,你得到了以下赎回信息:

Bank A: full cash value for customers with an account; otherwise 91.1% of your coins’ value.
::银行A:有账户的客户的全额现金价值;否则,占你硬币价值的91.1%。

Bank B: full cash value for customers with an account if the value is put on a merchant’s gift card that has use fees; otherwise 92% of your coin’s value.
::B银行:如果该价值被贴在有使用费的商家礼品卡上,客户可使用账户的全额现金价值;否则,你硬币价值的92%。

Bank C: full cash value for customers with an account if the value is $100 or less; otherwise 99% of the coins’ value.
::C银行:账户客户的全额现金价值,如果其价值为100美元或100美元以下;否则为硬币价值的99%。

When you told your friends about the redemption information, one of them said: “Oh, really! The money you’ll get for your coins is not a function of their actual value.” Was your friend correct? Why or why not?
::当你告诉你的朋友关于赎罪的信息的时候,他们中的一个人说:真的,你为你的硬币所能得到的钱,不是由它们的实际价值决定的。你的朋友是正确的吗?为什么或为什么不是呢?

Functions and the Vertical Line Test
::函数和垂直线测试

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

Situation 1:
::情况1:

You are selling raffle tickets for a school fundraiser. Each ticket costs $3.00
::你为学校募捐卖彩票每张票要三块

Situation 2:
::情况2:

You collect data from several students in your class on their ages and their heights: (18, 65″), (17, 64″), (18, 67″), (18, 68″), (17,66″)
::你从班级中的若干学生那里收集他们年龄和身高的数据: (18,65), (17, 64), (18, 67), (18, 68), (17,66)

In the first situation, let the variable $\begin{aligned} x \end{aligned}$ represent the number of raffle tickets that you sell, and let $\begin{aligned} y \end{aligned}$ represent the amount of money you make. If you sell $\begin{aligned} x \end{aligned}$ raffle tickets, you will make $\begin{aligned} y = 3 x \end{aligned}$ dollars; there is one and only one number representing your profit. Notice that you can use the number of raffle tickets you sell to predict how much money you will make. This is an example of a function.
::在第一种情况下,让变量 x 代表您出售的彩票数量,让您代表您赚的钱数量。如果您卖的是彩票数量,您就会赚到 Y=3x美元;只有一个数字代表您的利润。请注意,您可以使用你卖的彩票数量来预测您能赚多少钱。这是一个函数的例子。

Now consider the second situation. Can you similarly use the data to predict specific height, based on age? No, this is not the case in the second situation. For example, if a student is 18 years old, there are multiple heights that the student could be. This situation is not a function.
::现在考虑第二种情况。您能否同样使用这些数据来预测特定身高, 以年龄为基础? 不, 第二种情况不是这样。例如, 如果学生年满18岁, 学生可能身高多重。这种情况不是一个函数。

A function is a relationship between an independent variable , the input, and a dependent variable, the output, where each input value of the independent variable , corresponds to one and only one output value of the dependent variable .
::函数是独立变量、输入和依附变量之间的关系。输出,即独立变量的每个输入值对应独立变量的单一输出值。

It is important to note that both situations above are relations. A relation is simply a relationship between two sets of numbers or data. For example, in the second situation, we created a relationship between students’ ages and heights, just by writing each student’s information as an ordered pair .
::必须指出,上述两种情况都是关系。一种关系仅仅是两套数字或数据之间的关系。比如,在第二种情况下,我们在学生年龄和身高之间建立了关系,只是把每个学生的信息写成一对定做的。

Functions may be presented in many ways. Some of the most common ways to represent functions include: sets of ordered pairs (e.g., in a table), written or equation rules, and graphs.
::功能可以多种方式表现。一些最常见的功能代表方式包括:一对一对一对一对一对一对一对一对一对一对(如在表格中)、书面规则或方程规则、图表一对一对一对一对一对一对一对一对。

The notation used to show that there is a functional relationship between the is called functional notation. The typical notation for a function is $\begin{aligned} f (x) \end{aligned}$ , which is another way of representing the dependent variable $\begin{aligned} y \end{aligned}$ in an equation. The function $\begin{aligned} y = 3 x \end{aligned}$ would be represented using as follows:
::用于显示函数之间有功能关系的符号,称为功能符号。函数的典型符号是f(x),这是在方程式中代表依附变量y的另一种方式。函数 y=3x 将使用下列方式表示:

\begin{aligned} f (x) = 3 x \end{aligned}

fx)=3x

The table below shows the different ways to represent a relation:
::下表显示了反映某种关系的不同方式:

Representation ::代表权代表权代表权代表权代表权	Example ::示例示例示例示例
Set of ordered pairs ::一套定购一对	(1,3), (2,6), (3,9), (4,12) (a subset of the ordered pairs for this function) :这一职能的定购配对的分项)
Equation ::平方	$\begin{aligned} y = 3 x or f (x) = 3 x \end{aligned}$ ::y=3x 或 f( x)=3x
Graph ::图图图图图图图

In the first 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 first number in each pair (the $\begin{aligned} x \end{aligned}$ -value) is different, so we can be certain that there are no cases where 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配对。为了验证这是一个函数, 我们必须确保每个 x 值与一个 Y 值相关联。在此示例中, 每对的首个数( x 值) 不同, 因此我们可以确定没有特定 x 与一个 y 以上相关连的情况。

In the second 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 3.
::在第二个表示式,即线的等式中,很明显,以任何数字取代x将产生不同的y,因为x数字只是乘以3。

The third representation above is a graph. A quick and effective visual to decide if a graph is a function is by doing a “ vertical line test ”. If all possible vertical lines only cross the relation in one place, then the relation is a function. 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.
::以上第三个表示式是图表。一个快速有效的直观图像可以确定图形是否是一个函数,即进行“垂直线测试”。如果所有可能的垂直线只跨越一个地方的关系,那么关系就是一个函数。如果在图形的任何地方可以画一条垂直线,使该线跨越两个地方的关系,那么该关系就不是一个函数。

Vertical Line Test
::垂直线测试

A graphed relation is a function if there are no vertical lines that intersect the graphed relation in more than one point.
::如果没有垂直线将图形关系交叉到不止一个点,则图形关系是一个函数。

Are each of these graphed relations a function?
::这些图表显示的关系都是函数吗?

For graph 1, by drawing a vertical line (the red line) through the graph, we can see that the vertical line intersects the circle more than once. Therefore , this graph is NOT a function.
::对于图1,通过在图中绘制垂直线(红线),我们可以看到垂直线将圆交错不止一次。因此,这个图不是函数。

For graph 2, No matter where a vertical line is drawn through the graph, there will be only one intersection . Therefore, this graph is a function.
::对于图2,无论垂直线通过图中绘制,都将只有一个交叉点。因此,此图是一个函数。

Examples
::实例

Example 1
::例1

Earlier, you were asked about getting cash for a $250 stash of coins and if the data you received from the banks represent a function. Your friend thinks not but you are not sure.
::早些时候,有人问过你要用250美元的硬币存取现金,以及从银行收到的数据是否代表一种功能。你朋友不认为,但你不确定。

If we were to organize the information received from the banks into ordered pairs $\begin{aligned} (x, y) \end{aligned}$ , it might look something like:
::如果我们把从银行收到的信息整理成一对定单(x,y),

Bank A: ($250, $250), ($250, $250*0.911)
::银行A250,250美元),250美元,250美元*0.911美元

Bank B: ($250, $250-fees), ($250, $250*0.92)
::B银行:250美元、250美元(250美元)、250美元、250美元*0.92

Bank C: ($250, $250*0.99)
::C银行:250美元,250美元*0.99美元

Each $\begin{aligned} x \end{aligned}$ value, the independent variable, represents the real value of coins, in this case $250, and each $\begin{aligned} y \end{aligned}$ value, the dependent variable, represents the money the bank will give you for the coins.
::每个 x 值, 独立的变量, 代表硬币的真实价值, 在本案中为250美元, 而每个y 值, 依附变量, 代表银行会给你的硬币的金额。

Since there are many different $\begin{aligned} y \end{aligned}$ values for the one $\begin{aligned} x \end{aligned}$ value, the above relationships definitely are not a function. Your friend was right.
::由于一个 x 值有许多不同的y 值, 上述关系绝对不是一个函数。您的朋友是对的。

Example 2
::例2

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

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

Function defined as:
::函数定义如下:

(2, 4), (3, 9), (5, 11), (5, 12) is not a function because 5 is paired with 11 and with 12.
:2,4,3,9,5,11,5,12)不是一项职能,因为5与11和12相配。

The graph displaying a 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.
::显示关联的图形是一个函数,因为每个 xis 配对时只有 1 个。通过图形的垂直线将总是只遇到一个点。

For the following examples, determine if the relation is a function.
::对于下列例子,确定这种关系是否是一种功能。

Example 3
::例3

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

Don’t be fooled! This is a function, there is only one unique output for each input. The fact that both $\begin{aligned} x \end{aligned}$ values 2.1 and 1 are associated with $\begin{aligned} y \end{aligned}$ value 4 does not mean that 2.1 and 1 don’t have a specific associated value. Also, not matter how close two $\begin{aligned} x \end{aligned}$ '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, 每个输入只有一个独特的输出。 xvalue 2. 1 和 1 都与 yvalu 4 相关,这并不意味着 2.1 和 1 都没有具体的关联值。此外,无论两个 x 2 ( 比如 2 和 2. 1 ) 的距离有多近,如果它们不完全相同,它们都不会影响函数的定义。

Example 4
::例4

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

Any value chosen for $\begin{aligned} x \end{aligned}$ has one and only one associated value for $\begin{aligned} y \end{aligned}$ (4 times as big) and therefore is a function.
::为 xhas 所选择的 Y( 4 次大) 仅一个相关值的 xhas 1 值所选择的任何值,因此是一个函数。

Example 5
::例5

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

This is not a function. This graph looks like a “<”, with the point on the origin. Any value chosen for $\begin{aligned} x \end{aligned}$ will have 2 associated $\begin{aligned} y \end{aligned}$ values. For instance: $\begin{aligned} 4 = | - 4 | a n d 4 = | 4 | \end{aligned}$ .
::这不是函数。此图表看起来像一个“ < ” , 并带有源点。为 x 所选择的任何值将包含 2 个相关的 y 值。例如 : 4\ 4\ 4\ 和 4\ 4\ 4\ 。

Review
::回顾

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

Can a function definition be written in the form $\begin{aligned} x = 3 y \end{aligned}$ instead of $\begin{aligned} y = 3 x \end{aligned}$ ?
::函数定义能否以 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 #6-14, identify each relation as either a function, or not a function:
::对于#6-14,将每一关系确定为函数或非函数:

$\begin{aligned} (2, 4) (4, 6) (6, 8) (3, 4) (5, 7) (8, 2) \end{aligned}$
$\begin{aligned} (- 1, 6) (0, 4) (- 4, 0) (- 1, - 6) (- 3, - 8) \end{aligned}$

(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?
::在一次舞会上,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,在同一场舞蹈中,科里带着两个日期出现,这是否改变了答案?

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。

Section outline

Functions and the Vertical Line Test ::函数和垂直线测试

Vertical Line Test ::垂直线测试

Examples ::实例

Example 1 ::例1

Example 2 ::例2

For the following examples, determine if the relation is a function. ::对于下列例子,确定这种关系是否是一种功能。

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)