课程： 1.10 职能的域域和范围

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函数的域域和范围

Functions
::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

A function is a rule for relating two or more variables. For example, the price you pay for phone service may depend on the number of minutes you talk on the phone. We would say that the cost of phone service is a function of the number of minutes you talk. Consider the following situation:
::函数是连接两个或两个以上变量的规则。例如,您为电话服务支付的价格可能取决于您在电话中交谈的分钟数。我们会说,电话服务的费用取决于您交谈的分钟数。请考虑以下情况:

Josh goes to an amusement park where he pays $2 per ride.
::Josh去游乐园,他每骑一程要付2美元

There is a relationship between the number of rides Josh goes on and the total amount he spends that day: To figure out the amount he spends, we multiply the number of rides by two. This rule is an example of a function . Functions usually— but not always —are rules based on mathematical operations . You can think of a function as a box or a machine that contains a mathematical operation .
::Josh继续骑车的次数和他当天所花的总额之间有某种关系:为了弄清楚他所花的金额,我们将骑车次数乘以二。这是函数的一个例子。函数通常是基于数学操作的规则,但并不总是如此。你可以想象一个函数是装有数学操作的盒子或机器。

Whatever number we feed into the function box is changed by the given operation, and a new number comes out the other side of the box. When we input different values for the number of rides Josh goes on, we get different values for the amount of money he spends.
::无论我们输入到函数框中的数字因给定操作而改变,新的数字会从框的另一面出来。当我们输入不同数值计算乘车次数时,Josh继续输入,我们得到不同的数值计算他所花的钱数。

The input is called the independent variable because its value can be any number. The output is called the dependent variable because its value depends on the input value.
::输入被称为独立变量, 因为它的值可以是任意数字。输出被称为依附变量, 因为它的值取决于输入值。

Functions usually contain more than one mathematical operation. Here is a situation that is slightly more complicated than the example above:
::函数通常包含一个以上的数学操作。这里的情况比以上示例略为复杂 :

Jason goes to an amusement park where he pays $8 admission and $2 per ride.
::杰森去一个游乐公园在那里他支付8美元入场费和每程2美元。

The following function represents the total amount Jason pays. The rule for this function is "multiply the number of rides by 2 and add 8."
::以下函数表示Jason支付的总金额。此函数的规则是“乘数乘以2再加8。”

When we input different values for the number of rides, we arrive at different outputs (costs).
::当我们输入乘车次数的不同值时,我们得出了不同的产出(成本)。

These flow diagrams are useful in visualizing what a function is. However, they are cumbersome to use in practice. In algebra, we use the following short-hand notation instead:
::这些流程图有助于直观地显示函数是什么。但是,它们在实践中使用起来很麻烦。在代数中,我们使用以下的简称:

\begin{aligned} input \\ ↓ \\ \underset{⏟}{f (x)} & = y \leftarrow output \\ function \\ box \end{aligned}

::输入 f( x) 输出函数框

First, we define the variables:
::首先,我们定义变量:

$\begin{aligned} x = \end{aligned}$ the number of rides Jason goes on
::x= 乘车次数贾森继续

$\begin{aligned} y = \end{aligned}$ the total amount of money Jason spends at the amusement park.
::杰森花在游乐园的钱总额

So, $\begin{aligned} x \end{aligned}$ represents the input and $\begin{aligned} y \end{aligned}$ represents the output. The notation $\begin{aligned} f () \end{aligned}$ represents the function or the mathematical operations we use on the input to get the output. In the last example, the cost is 2 times the number of rides plus 8. This can be written as a function:
::所以, x 代表输入, y 代表输出。 f () 符号表示输入时我们用来获取输出的函数或数学操作。在最后一个例子中, 成本是乘数的2倍 + 8 。这可以写为函数 :

\begin{aligned} f (x) = 2 x + 8 \end{aligned}

xx)=2x+8

In algebra, the notations $\begin{aligned} y \end{aligned}$ and $\begin{aligned} f (x) \end{aligned}$ are typically used interchangeably. Technically, though, $\begin{aligned} f (x) \end{aligned}$ represents the function itself and $\begin{aligned} y \end{aligned}$ represents the output of the function.
::在代数中,符号y和f(x)通常可以互换使用。但从技术上讲,f(x)代表函数本身,y代表函数的输出。

Identify the Domain and Range of a Function
::指定函数的域域和范围

In the last example, we saw that we can input the number of rides into the function to give us the total cost for going to the amusement park. The set of all values that we can use for the input is called the domain of the function, and the set of all values that the output could turn out to be is called the range of the function. In many situations the domain and range of a function are both simply the set of all real numbers, but this isn’t always the case. Let's look at our amusement park example.
::在最后一个例子中,我们看到,我们可以将骑车的数量输入到这个函数中,以便给我们去游乐园的总成本。我们可以用于输入的所有数值都被称为函数的域,而产出可以被证实的所有数值的组则被称为函数的范围。在许多情况下,一个函数的域和范围都是所有实际数字的组,但并不总是这样。让我们看看我们的游乐园的例子。

Finding the Domain and Range of a Function
::查找函数的域域和范围

1. Find the of the function that describes the situation:
::1. 确定描述情况的职能:

Jason goes to an amusement park where he pays $8 admission and $2 per ride.
::杰森去一个游乐公园在那里他支付8美元入场费和每程2美元。

Here is the function that describes this situation:
::以下是描述这种情况的函数:

\begin{aligned} f (x) = 2 x + 8 = y \end{aligned}

::f(x)=2x+8=y

In this function, $\begin{aligned} x \end{aligned}$ is the number of rides and $\begin{aligned} y \end{aligned}$ is the total cost. To find the domain of the function, we need to determine which numbers make sense to use as the input $\begin{aligned} (x) \end{aligned}$ .
::3⁄4 ̄ ̧漯B

The values have to be zero or positive, because Jason can't go on a negative number of rides.
::数值必须是零或正数, 因为杰森不能去负数骑车。

The values have to be integers because, for example, Jason could not go on 2.25 rides.
::数值必须是整数,因为,例如,杰森不能乘2.25乘飞机。

Realistically, there must be a maximum number of rides that Jason can go on because the park closes, he runs out of money, etc. However, since we aren’t given any information about what that maximum might be, we must consider that all non-negative integers are possible values regardless of how big they are.
::现实地说,Jason必须拥有最多数量的旅行,因为公园关闭,他没有钱等等。然而,由于我们没有得到任何关于最高数量的信息,我们必须考虑所有非负整数都是可能的值,不管它们有多大。

For this function, the domain is the set of all non-negative integers.
::对于此函数,域是所有非负整数的一组。

To find the range of the function we must determine what the values of $\begin{aligned} y \end{aligned}$ will be when we apply the function to the input values. The domain is the set of all non-negative integers: {0, 1, 2, 3, 4, 5, 6, ...}. Next we plug these values into the function for $\begin{aligned} x \end{aligned}$ . If we plug in 0, we get 8; if we plug in 1, we get 10; if we plug in 2, we get 12, and so on, counting by 2s each time. Possible values of $\begin{aligned} y \end{aligned}$ are therefore 8, 10, 12, 14, 16, 18, 20... or in other words all even integers greater than or equal to 8.
::要找到函数范围, 我们必须在将函数应用到输入值时确定 y 的值。域是所有非负整数的集合 : {0, 1, 2, 3, 4, 4, 5, 6, ......} 。下一步我们将这些值插入 x 的函数中。如果我们插入 0, 我们得到 8; 如果我们插入 1, 我们得到 10; 如果我们插入 2, 我们得到 10; 如果我们插入 2, 我们得到 12, 如此, 每次计算 2。 y 可能值为 8, 10, 12, 14, 16, 18, 20... , 或换句话说, 所有的整数都大于或等于 8 。

The range of this function is the set of all even integers greater than or equal to 8.
::此函数的范围是所有大于或等于8的偶数整数的集合。

2. Find the domain and range of the following functions.
::2. 确定下列职能的域和范围。

a) A ball is dropped from a height and it bounces up to 75% of its original height.
:a) 球从高处投下,弹出至其原高度的75%。

Let’s define the variables:
::让我们定义变量:

$\begin{aligned} x = \end{aligned}$ original height
::x= 原始高度

$\begin{aligned} y = \end{aligned}$ bounce height
::Y=弹跳高度

A function that describes the situation is $\begin{aligned} y = f (x) = 0.75 x \end{aligned}$ . $\begin{aligned} x \end{aligned}$ can represent any real value greater than zero, since you can drop a ball from any height greater than zero. A little thought tells us that $\begin{aligned} y \end{aligned}$ can also represent any real value greater than zero.
::函数 y=f(x) = 0.75x. x 表示任何实际值大于零,因为您可以从任何高度投球大于零。略微思考告诉我们,y 也可以代表任何实际值大于零。

The domain is the set of all real numbers greater than zero. The range is also the set of all real numbers greater than zero.
::域是所有实际数字大于零的一组。范围也是所有实际数字大于零的一组。

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

Since there is no word problem attached to this equation , we can assume that we can use any real number as a value of $\begin{aligned} x \end{aligned}$ . When we square a real number, we always get a non-negative answer, so $\begin{aligned} y \end{aligned}$ can be any non-negative real number.
::由于这个等式没有字的问题, 我们可以假设我们可以用任何实际数字作为x值。当我们平方一个实际数字时, 我们总是得到一个非负的答案, y 可以是任何非负的真实数字。

The domain of this function is all real numbers. The range of this function is all non-negative real numbers.
::此函数的域是所有实际数字。此函数的范围是所有非负实际数字。

In the functions we’ve looked at so far, $\begin{aligned} x \end{aligned}$ is called the independent variable because it can be any of the values from the domain, and $\begin{aligned} y \end{aligned}$ is called the dependent variable because its value depends on $\begin{aligned} x \end{aligned}$ . However, any letters or symbols can be used to represent the dependent and independent variables. Here are three different examples:
::在我们迄今所查看的函数中, x 被称为独立变量, 因为它可以是域中的任何值, y 则被称为依附变量, 因为它的值取决于 x。但是, 任何字母或符号都可以用来代表依附和独立的变量。下面有三个不同的例子:

\begin{aligned} y = f (x) = 3 x \\ R = f (w) = 3 w \\ v = f (t) = 3 t \end{aligned}

::y=f(x)=3xR=f(w)=3wv=f(t)=3t

These expressions all represent the same function: a function where the dependent variable is three times the independent variable. Only the symbols are different. In practice, we usually pick symbols for the dependent and independent variables based on what they represent in the real world—like $\begin{aligned} t \end{aligned}$ for time, $\begin{aligned} d \end{aligned}$ for distance , $\begin{aligned} v \end{aligned}$ for velocity, and so on. But when the variables don’t represent anything in the real world—or even sometimes when they do—we traditionally use $\begin{aligned} y \end{aligned}$ for the dependent variable and $\begin{aligned} x \end{aligned}$ for the independent variable.
::这些表达式都代表着相同的函数:一个函数,其依附变量是独立的变量的三倍。只有符号是不同的。在实践中,我们通常根据在现实世界中所代表的依附变量和独立变量的符号来选择这些变量的符号 — — 比如时间的 t,距离的 d,速度的 v,等等。但是当变量在现实世界中并不代表任何东西时 — — 甚至是有时,我们通常使用y来表示依附变量和独立变量的 x。

Make a Table For a Function
::设置函数的表格

A table is a very useful way of arranging the data represented by a function. We can match the input and output values and arrange them as a table. For example, the values from Example 1 above can be arranged in a table as follows:
::表格是安排函数所代表数据的一种非常有用的方式。我们可以将输入值和输出值匹配,并将它们排列为表格。例如,上文例1中的数值可以排列在以下表格中:

\begin{aligned} x : 0 1 2 3 4 5 6 \\ y : 8 10 12 14 16 18 20 \end{aligned}

::x:01 2 3 4 4 5 5 6 6y:81012141820

A table lets us organize our data in a compact manner. It also provides an easy reference for looking up data, and it gives us a set of coordinate points that we can plot to create a graph of the function.
::表格让我们能够以紧凑的方式组织我们的数据。它也为查找数据提供了方便的参考,它为我们提供了一组协调点,我们可以绘制一个函数图。

Constructing a Table of Values
::构建值表

Make a table of values for the function $\begin{aligned} f (x) = \frac{1}{x} \end{aligned}$ . Use the following numbers for input values: -1, -0.5, -0.2, -0.1, -0.01, 0.01, 0.1, 0.2, 0.5, 1.
::函数 f( x) = 1x 的数值表。输入值使用以下数字: -1, -0.5, -0. 0.2, -0.1, -0.01, 0.01, 0.01, 0.1, 0.1, 0.2, 0.2, 0.2, -0.1, 0.5, 0.5, 1。

Make a table of values by filling the first row with the input values and the next row with the output values calculated using the given function.
::将输入值填入第一行,然后用使用给定函数计算的输出值填入数值表,然后用下一个行填入数值表。

$\begin{aligned} x \end{aligned}$	-1	-0.5	-0.2	-0.1	-0.01	0.01	0.1	0.2	0.5	1
$\begin{aligned} f (x) = \frac{1}{x} \end{aligned}$	$\begin{aligned} \frac{1}{-1} \end{aligned}$	$\begin{aligned} \frac{1}{-0.5} \end{aligned}$	$\begin{aligned} \frac{1}{-0.2} \end{aligned}$	$\begin{aligned} \frac{1}{-0.1} \end{aligned}$	$\begin{aligned} \frac{1}{-0.01} \end{aligned}$	$\begin{aligned} \frac{1}{0.01} \end{aligned}$	$\begin{aligned} \frac{1}{0.1} \end{aligned}$	$\begin{aligned} \frac{1}{0.2} \end{aligned}$	$\begin{aligned} \frac{1}{0.5} \end{aligned}$	$\begin{aligned} \frac{1}{1} \end{aligned}$
$\begin{aligned} y \end{aligned}$	-1	-2	-5	-10	-100	100	10	5	2	1

When you’re given a function, you won’t usually be told what input values to use; you’ll need to decide for yourself what values to pick based on what kind of function you’re dealing with. We will discuss how to pick input values throughout these lessons.
::当您被赋予一个函数时, 您通常不会被告知要使用什么输入值; 您需要自己决定根据您正在处理的哪种功能选择哪些值。我们将讨论如何在这些课程中选择输入值。

Example
::示例示例示例示例

Example 1
::例1

Identify the domain and then make a table of values for the function $\begin{aligned} f (x) = \frac{1}{\sqrt{x}} \end{aligned}$ . Use the following numbers for input values: 0.01, 0.16, 0.25, 1, 4.
::识别域, 然后为函数 f( x) = 1x 绘制一个数值表。输入值使用以下数字: 0.01, 0.16, 0. 25, 1, 4。

Since you cannot compute the square root of negative numbers, these cannot be in the domain. Since we cannot have 0 in the denominator, 0 is also not in the domain. This means that the domain is all real numbers greater than zero.
::由于您无法计算负数的平方根, 这些无法在域内。由于分母中不能有 0, 0 也不属于域内。这意味着域内的所有实际数字都大于零。

$\begin{aligned} x \end{aligned}$	0.01	0.16	0.25	1	4
$\begin{aligned} f (x) = \frac{1}{x} \end{aligned}$	$\begin{aligned} \frac{1}{\sqrt{0.01}} \end{aligned}$	$\begin{aligned} \frac{1}{\sqrt{0.16}} \end{aligned}$	$\begin{aligned} \frac{1}{\sqrt{0.25}} \end{aligned}$	$\begin{aligned} \frac{1}{\sqrt{1}} \end{aligned}$	$\begin{aligned} \frac{1}{\sqrt{4}} \end{aligned}$
$\begin{aligned} y \end{aligned}$	10	2.5	2	1	0.5

Review
::回顾

For 1-6, identify the domain and range of the following functions.
::1-6,确定下列职能的领域和范围。

Dustin charges $10 per hour for mowing lawns.
::Dustin每小时为修剪草坪收费10美元。

Maria charges $25 per hour for tutoring math, with a minimum charge of $15.
::Maria每小时25美元的数学辅导费,最低收费为15美元。

$\begin{aligned} f (x) = 15 x - 12 \end{aligned}$
:xx)=15x-12)

$\begin{aligned} f (x) = 2 x^{2} + 5 \end{aligned}$
:xx)=2x2+5

$\begin{aligned} f (x) = \frac{1}{x} \end{aligned}$
:xx)=1x

$\begin{aligned} f (x) = \sqrt[3]{x} \end{aligned}$
:xx)=x3

What is the range of the function $\begin{aligned} y = x^{2} - 5 \end{aligned}$ when the domain is -2, -1, 0, 1, 2?
::当域为 -2, -1, 0, 1, 2 时, y=x2 - 5 函数的范围是多少?

What is the range of the function $\begin{aligned} y = 2 x - \frac{3}{4} \end{aligned}$ when the domain is -2.5, -1.5, 5?
::当域为 -2.5, -1.5, 5 时, y=2x-34 函数的范围是什么 ?

What is the domain of the function $\begin{aligned} y = 3 x \end{aligned}$ when the range is 9, 12, 15?
::当范围为 9、 12、 15 时, y=3x 函数的域是什么 ?

What is the range of the function $\begin{aligned} y = 3 x \end{aligned}$ when the domain is 9, 12, 15?
::当域为 9、 12、 15 时, y=3x 函数的范围是什么 ?

Angie makes $6.50 per hour working as a cashier at the grocery store. Make a table that shows how much she earns if she works 5, 10, 15, 20, 25, or 30 hours.
::安琪在杂货店当收银员每小时挣6.50美元。制作一张表格,显示如果工作5、10、15、20、25或30小时,她挣多少钱。

The area of a triangle is given by the formula $\begin{aligned} A = \frac{1}{2} b h \end{aligned}$ . If the base of the triangle measures 8 centimeters, make a table that shows the area of the triangle for heights 1, 2, 3, 4, 5, and 6 centimeters.
::A=12bh 公式给出三角形区域。如果三角形的底部为8厘米,请绘制一个表格,显示高1、2、3、4、5和6厘米的三角形区域。

Make a table of values for the function $\begin{aligned} f (x) = \sqrt{2} x + 3 \end{aligned}$ for input values -1, 0, 1, 2, 3, 4, 5.
::为输入值 -1, 0, 1, 2, 3, 4, 5 的 f(x) =2x+3 函数的数值制成表格。

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

常规

函数的域域和范围

Functions ::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

Identify the Domain and Range of a Function ::指定函数的域域和范围

Finding the Domain and Range of a Function ::查找函数的域域和范围

Make a Table For a Function ::设置函数的表格

Constructing a Table of Values ::构建值表

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)