Section: 《书写功能规则》 | 9.3 书面职能规则

Section outline

Alfred is getting ready to decorate the yard for Halloween. He always lights both sides of the driveway with carved jack-o-lanterns. Alfred knows that Jacob, a local farmer sells his smaller pumpkins for $2.75 each. He needs to write some type of a function rule to determine the cost of 25, 35, 45 and 50 pumpkins. How can Alfred use the information he knows to write the function rule?
::Alfred正准备为万圣节装饰院子。他总是用刻刻的胡萝卜点亮车道两侧。 Alfred知道,当地农民Jacob每人售出小南瓜2.75美元。他需要写出某种功能规则来确定25、35、45和50南瓜的成本。Alfred如何使用他所知道的信息来写功能规则?

In this concept, you will learn to write function rules.
::在这个概念中,你会学会写出功能规则。

Function Rule
::职能规则

A function rule expresses the output value in terms of all operations performed on the input value. A function rule can be expressed either using words or symbols. When a function rule is written from the information given in an input/output table, the pattern represented in the table must be represented by the function rule.
::函数规则函数规则表示根据输入值进行的所有操作的输出值。函数规则可以用文字或符号表示。当函数规则是从输入/产出表中提供的信息中写入时, 表格中显示的模式必须用函数规则表示。

Let’s write a function rule for each of the following input/output tables:
::让我们为以下每个输入/产出表格写一个函数规则:

Input ( $\begin{aligned} x \end{aligned}$ )	Output ( $\begin{aligned} y \end{aligned}$ )
2	7
3	10
4	13
5	16

First, look for a pattern between the input numbers and the output numbers.
::首先,在输入数字和输出数字之间寻找一个模式。

The values of the output numbers are very close to the input number times three.
::输出数字的值非常接近输入数乘以 3 。

\begin{aligned} y = 3 x \end{aligned}

::y=3x y=3x

Next, try the function rule to see if it works.
::接下来,试一下函数规则看看它是否有效。

\begin{aligned} \begin{array}{rcl} y & = & 3 x \\ y & = & 3 (2) \\ y & = & 6 \end{array} \end{aligned}

::y=3xy=3(2)y=6 y=6

\begin{aligned} \begin{array}{rcl} y & = & 3 x \\ y & = & 3 (3) \\ y & = & 9 \end{array} \end{aligned}

::y=3xy=3(3)y=9 y=9

\begin{aligned} \begin{array}{rcl} y & = & 3 x \\ y & = & 3 (4) \\ y & = & 12 \end{array} \end{aligned}

::y=3xy=3(4)y=12

\begin{aligned} \begin{array}{rcl} y & = & 3 x \\ y & = & 3 (5) \\ y & = & 15 \end{array} \end{aligned}

::y=3xy=3(5)y=15

Next, if the function rule does not give the output number, determine what value could be added or subtracted from the original function rule to give the desired output number.
::其次,如果函数规则没有给出产出编号,则决定从原始函数规则中可以添加或减去什么值,以给出理想产出数字。

Each of the given output numbers is one greater than the output values found by $\begin{aligned} y = 3 x \end{aligned}$ .
::给定输出数字中的每个输出数字都大于y=3x所发现的输出值之一。

Next, write a new function rule to satisfy the values given in the table.
::下一步,写入一个新的函数规则,以满足表格中的值。

\begin{aligned} y = 3 x + 1 \end{aligned}

::y=3x+1 y=3x+1

Next, try the new function rule to see if it works.
::下一步, 尝试新的函数规则, 看看它是否有效。

\begin{aligned} \begin{array}{rcl} y & = & 3 x + 1 \\ y & = & 3 (2) + 1 \\ y & = & 6 + 1 \\ y & = & 7 \end{array} \end{aligned}

::y=3x+1y=3(2)+1y=6+1y=7

\begin{aligned} \begin{array}{rcl} y & = & 3 x + 1 \\ y & = & 3 (3) + 1 \\ y & = & 9 + 1 \\ y & = & 10 \end{array} \end{aligned}

::y=3x+1y=3(3)+1y=9+1y=10

\begin{aligned} \begin{array}{rcl} y & = & 3 x + 1 \\ y & = & 3 (4) + 1 \\ y & = & 12 + 1 \\ y & = & 13 \end{array} \end{aligned}

::y=3x+1y=3(4)+1y=12+1y=13

\begin{aligned} \begin{array}{rcl} y & = & 3 x + 1 \\ y & = & 3 (5) + 1 \\ y & = & 15 + 1 \\ y & = & 16 \end{array} \end{aligned}

::y=3x+1y=3(5)+1y=15+1y=16

The function rule $\begin{aligned} y = 3 x + 1 \end{aligned}$ gives the output values shown in the table when the given input values are used.
::函数规则 y=3x+1 在使用给定输入值时给出表格中显示的输出值。

Input ( $\begin{aligned} x \end{aligned}$ )	Output ( $\begin{aligned} y \end{aligned}$ )
-3	-2
-1	2
1	6
3	10

First, look for a pattern between the input numbers and the output numbers.
::首先,在输入数字和输出数字之间寻找一个模式。

The pattern between the values of the output numbers and the input number is not obvious. Perhaps trial and error may work to get started. Begin by doubling each of the input numbers.
::输出数值和输入数值之间的模式并不明显。也许尝试和错误可以启动。开始将每个输入数翻一番。

\begin{aligned} y = 2 x \end{aligned}

::y=2x y=2x

Next, try the function rule to see if it works.
::接下来,试一下函数规则看看它是否有效。

\begin{aligned} \begin{array}{rcl} y & = & 2 x \\ y & = & 2 (- 3) \\ y & = & - 6 \end{array} \end{aligned}

::y=2xy=2(- 3)y6

\begin{aligned} \begin{array}{rcl} y & = & 2 x \\ y & = & 2 (- 1) \\ y & = & - 2 \end{array} \end{aligned}

::y= 2xy= 2 (2- 1) y @%% 2

\begin{aligned} \begin{array}{rcl} y & = & 2 x \\ y & = & 2 (1) \\ y & = & 2 \end{array} \end{aligned}

::y=2xy=2(1)y=2

\begin{aligned} \begin{array}{rcl} y & = & 2 x \\ y & = & 2 (3) \\ y & = & 6 \end{array} \end{aligned}

::y=2xy=2(3)y=6 y=6

When $\begin{aligned} x = 1 \end{aligned}$ in the function rule $\begin{aligned} y = 2 x \end{aligned}$ , the output number was $\begin{aligned} y = 2 \end{aligned}$ . If 4 were added to this, then the value of ‘ $\begin{aligned} y \end{aligned}$ ’ would be 6 which is the given output number in the table.
::当函数规则y=2x中的 x=1 时,输出号为 y=2。如果在此添加 4, 则“y” 值为 6, 即表格中给定的输出号。

Next, write a new function rule to satisfy the values given in the table.
::下一步,写入一个新的函数规则,以满足表格中的值。

\begin{aligned} y = 2 x + 4 \end{aligned}

::y=2x+4 y=2x+4

Next, try the new function rule to see if it works.
::下一步, 尝试新的函数规则, 看看它是否有效。

\begin{aligned} \begin{array}{rcl} x & = & - 3 \\ y & = & 2 x + 4 \\ y & = & 2 (- 3) + 4 \\ y & = & - 6 + 4 \\ y & = & - 2 \end{array} \end{aligned}

::x3y=2x+4y=2(-3)+4y=6+4y=2

\begin{aligned} \begin{array}{rcl} x & = & - 1 \\ y & = & 2 x + 4 \\ y & = & 2 (- 1) + 4 \\ y & = & - 2 + 4 \\ y & = & 2 \end{array} \end{aligned}

::x1y=2x+4y=2(-1)+4y}2+4y=2

\begin{aligned} \begin{array}{rcl} x & = & 1 \\ y & = & 2 x + 4 \\ y & = & 2 (1) + 4 \\ y & = & 2 + 4 \\ y & = & 6 \end{array} \end{aligned}

::x=1y=2x+4y=2(1)+4y=2+4y=2+4y=6

\begin{aligned} \begin{array}{rcl} x & = & 3 \\ y & = & 2 x + 4 \\ y & = & 2 (3) + 4 \\ y & = & 6 + 4 \\ y & = & 10 \end{array} \end{aligned}

::x=3y=2x+4y=2(3)+4y=6+4y=10

The function rule $\begin{aligned} y = 2 x + 4 \end{aligned}$ gives the output values shown in the table when the given input values are used.
::函数规则 y=2x+4 在使用给定输入值时给出表格中显示的输出值。

Input ( $\begin{aligned} x \end{aligned}$ )	Output ( $\begin{aligned} y \end{aligned}$ )
2	7
3	10
4	13
5	16

The above input/output table can also be written as
::上述投入/产出表也可以作为

$\begin{aligned} x \end{aligned}$	$\begin{aligned} f (x) \end{aligned}$
2	7
3	10
4	13
5	16

The input values are now represented by ‘ $\begin{aligned} x \end{aligned}$ ’ and the output numbers ‘ $\begin{aligned} y \end{aligned}$ ’ are represented by $\begin{aligned} f (x) \end{aligned}$ .
::输入值现在以 'x ' 表示,输出数字`y ' 以 f(x)表示。

Therefore , the function rule $\begin{aligned} y = 3 x + 1 \end{aligned}$ can also be written as $\begin{aligned} f (x) = 3 x + 1 \end{aligned}$ . This way of writing a function rule is known as .
::因此,函数规则 y=3x+1 也可以写成 f(x)=3x+1 。这种写函数规则的方式被称为。

\begin{aligned} f (x) = 3 x + 1 \end{aligned}

xx)=3x+1

The above function notation can be read as:
::上述函数符号可读作:

There is a function ‘ $\begin{aligned} f \end{aligned}$ ’ written in terms of the variable ‘ $\begin{aligned} x \end{aligned}$ ’ such that it equals $\begin{aligned} 3 x + 1 \end{aligned}$ or $\begin{aligned} f \end{aligned}$ of $\begin{aligned} x \end{aligned}$ equals $\begin{aligned} 3 x + 1 \end{aligned}$ .
::以变量 'x ' 写成的函数 'f ' 等于 3x+1 或 x 的 f 等于 3x+1 或 f 等于 3x+1 。

If a function written in function notation form were given then to find the value of the function for the input value ‘3’ the function notation could be written as.
::如果以函数符号格式写入一个函数, 以找到输入值 '3' 的函数值, 函数符号可以写成。

If $\begin{aligned} f (x) = - 3 x + 5 \end{aligned}$ find $\begin{aligned} f (3) \end{aligned}$ .
::如果f(x)%%%3x+5 找到 f(3) 。

First, substitute ‘3’ for the value of ‘ $\begin{aligned} x \end{aligned}$ .’
::首先,将`x. ' 值改为`3 ' 。

\begin{aligned} \begin{array}{rcl} f (x) & = & - 3 x + 5 \\ f (3) & = & - 3 (3) + 5 \end{array} \end{aligned}

::f(x) 3x+5f(3) 3(3)+5

Next, perform the multiplication to clear the parenthesis.
::下一步,执行乘法清除括号。

\begin{aligned} \begin{array}{rcl} f (3) & = & - 3 (3) + 5 \\ f (3) & = & - 9 + 5 \end{array} \end{aligned}

::f(3)3(3)+5f(3)+9+5

Then, perform the addition on the right side of the function.
::然后在函数的右侧执行添加。

\begin{aligned} \begin{array}{rcl} f (3) & = & - 9 + 5 \\ f (3) & = & - 4 \end{array} \end{aligned}

::f(3)9+5f(3)4

The answer is -4.
::答案是 -4

If $\begin{aligned} f (x) = - 3 x + 5 \end{aligned}$ then $\begin{aligned} f (3) = - 4 \end{aligned}$ .
::如果f(x)3x+5,然后f(3)4。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Alfred and the pumpkins.
::之前,你得到一个问题关于阿尔弗雷德和南瓜。

Alfred needs to buy small pumpkins that cost $2.75 each. Alfred needs to write a function rule to figure out the cost of 25, 35, 45 and 50 pumpkins. What function rule can Alfred write using function notation?
::Alfred需要买小南瓜,每个南瓜花费2.75美元。Alfred需要写一个功能规则来计算25、35、45和50南瓜的成本。Alfred用功能符号写什么功能规则?

First, he can create a table to represent the information from the problem.
::首先,他可以创建一个表来代表来自问题的信息。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} f (x) \end{aligned}$
25
35
45
60

Next, write a function rule, using function notation that can be used to complete the table.
::下一步,写入函数规则,使用可用于填写表格的函数符号。

\begin{aligned} f (x) = 2.75 x \end{aligned}

fx)=2.75x

Next, use the function rule to determine the value of $\begin{aligned} f (x) \end{aligned}$ for each of the given ‘ $\begin{aligned} x \end{aligned}$ ’ values.
::下一步,使用函数规则来确定每个给定的 " x " 值的f(x)值。

If $\begin{aligned} f (x) = 2.75 x \end{aligned}$ find $\begin{aligned} f (25) \end{aligned}$ .
::如果f(x)=2.75x发现f(25).

Next, substitute 25 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::接下来,在函数规则中,以25取代`x ' 。

\begin{aligned} \begin{array}{rcl} f (x) & = & 2.75 x \\ f (25) & = & 2.75 (25) \end{array} \end{aligned}

xx) = 2.75xf(25) = 2.75(25)

Then, perform the multiplication on the right side of the function rule.
::然后在函数规则右侧执行乘法。

\begin{aligned} \begin{array}{rcl} f (25) & = & 2.75 (25) \\ f (25) & = & $ 68.75 \end{array} \end{aligned}

::f(25)=2.75(25)f(25)=68.75美元

The answer is 68.75.
::答案是68.75

Twenty-five pumpkins will cost $68.75.
::25个南瓜要花68.75美元

If $\begin{aligned} f (x) = 2.75 x \end{aligned}$ find $\begin{aligned} f (35) \end{aligned}$ .
::如果f(x)=2.75x发现f(35).

Next, substitute 35 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::接下来,在函数规则中,以35取代`x ' 。

\begin{aligned} \begin{array}{rcl} f (x) & = & 2.75 x \\ f (35) & = & 2.75 (35) \end{array} \end{aligned}

xx) = 2.75xf(35) = 2.75(35)

Then, perform the multiplication on the right side of the function rule.
::然后在函数规则右侧执行乘法。

\begin{aligned} \begin{array}{rcl} f (35) & = & 2.75 (35) \\ f (35) & = & $ 96.25 \end{array} \end{aligned}

::f(35)=2.75(35)f(35)=96.25美元

The answer is 96.25.
::答案是96.25

Thirty-five pumpkins will cost $96.25.
::35南南瓜将耗资96.25美元。

If $\begin{aligned} f (x) = 2.75 x \end{aligned}$ find $\begin{aligned} f (45) \end{aligned}$ .
::如果f(x)=2.75x发现f(45).

Next, substitute 45 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::接下来,在函数规则中以 " x " 代替45。

\begin{aligned} \begin{array}{rcl} f (x) & = & 2.75 x \\ f (45) & = & 2.75 (45) \end{array} \end{aligned}

xx)=2.75xf(45)=2.75(45)

Then, perform the multiplication on the right side of the function rule.
::然后在函数规则右侧执行乘法。

\begin{aligned} \begin{array}{rcl} f (45) & = & 2.75 (45) \\ f (45) & = & $ 123.75 \end{array} \end{aligned}

::f(45)=2.75(45)f(45)=123.75美元

The answer is 123.75.
::答案是123.75。

Forty-five pumpkins will cost $123.75.
::45个南瓜将花费123.75美元。

If $\begin{aligned} f (x) = 2.75 x \end{aligned}$ find $\begin{aligned} f (50) \end{aligned}$ .
::如果f(x)=2.75x发现f(50).

Next, substitute 50 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::接下来,在函数规则中以 " x " 代替50。

\begin{aligned} \begin{array}{rcl} f (x) & = & 2.75 x \\ f (50) & = & 2.75 (50) \end{array} \end{aligned}

xx)=2.75xf(50)=2.75(50)

Then, perform the multiplication on the right side of the function rule.
::然后在函数规则右侧执行乘法。

\begin{aligned} \begin{array}{rcl} f (50) & = & 2.75 (50) \\ f (50) & = & $ 137.50 \end{array} \end{aligned}

::f(50)=2.75(50)=137.50美元

The answer is 137.50.
::答案是137.50。

Fifty pumpkins will cost $137.50.
::50个南瓜将花费137.50美元。

Example 2
::例2

For the given table, write a function rule using function notation to represent the information given in the table.
::对于给定的表格,请写入函数规则,使用函数符号表示表格中的信息。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} f (x) \end{aligned}$
9	8
11	10
15	14
17	16

Remember, the input numbers have been replaced with ‘ $\begin{aligned} x \end{aligned}$ ’ and the output numbers have been replaced with $\begin{aligned} f (x) \end{aligned}$ .
::记住, 输入号已被 `x' 替换, 输出号已被 f(x) 替换。

First, look at the two columns of given numbers to determine a pattern between the values.
::首先,看看给定数字的两列,以确定数值之间的模式。

The numbers in the second column are one less than those in the first column.
::第二列的数字比第一列的数字少一个。

Next, write a function rule, using function notation, to represent the pattern.
::下一步,写入函数规则,使用函数符号表示模式。

\begin{aligned} f (x) = x - 1 \end{aligned}

xx)=x-1

Next, test the function rule for each of the given ‘ $\begin{aligned} x \end{aligned}$ ’ values.
::下一步,测试给定的每个 " x " 值的函数规则。

If $\begin{aligned} f (x) = x - 1 \end{aligned}$ find $\begin{aligned} f (9) \end{aligned}$ .
::如果f(x) =x-1 发现f(9) 。

Next, substitute 9 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::接下来,在函数规则中,将 " x " 改为 " 9 " 。

\begin{aligned} \begin{array}{rcl} f (x) & = & x - 1 \\ f (9) & = & 9 - 1 \end{array} \end{aligned}

xx) =x-1f(9) 9=9-1

Then, perform the subtraction on the right side of the function rule.
::然后在函数规则右侧执行减法。

\begin{aligned} \begin{array}{rcl} f (9) & = & 9 - 1 \\ f (9) & = & 8 \end{array} \end{aligned}

::f(9)=9-1f(9)=8

The answer is 8.
::答案是8岁

If $\begin{aligned} f (x) = x - 1 \end{aligned}$ find $\begin{aligned} f (11) \end{aligned}$ .
::如果f(x) =x-1 发现f(11) 。

Next, substitute 11 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::接下来,在函数规则中,以 " x " 代替11。

\begin{aligned} \begin{array}{rcl} f (x) & = & x - 1 \\ f (11) & = & 11 - 1 \end{array} \end{aligned}

xx) =x-1f(11) =11-1

Then, perform the subtraction on the right side of the function rule.
::然后在函数规则右侧执行减法。

\begin{aligned} \begin{array}{rcl} f (11) & = & 11 - 1 \\ f (11) & = & 10 \end{array} \end{aligned}

::f( 11) =11-1f( 11) =10

The answer is 10.
::答案是10个

If $\begin{aligned} f (x) = x - 1 \end{aligned}$ find $\begin{aligned} f (15) \end{aligned}$ .
::如果f(x) =x-1 发现f(15) 。

Next, substitute 15 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::接下来,在函数规则中,以15取代 " x " 。

\begin{aligned} \begin{array}{rcl} f (x) & = & x - 1 \\ f (15) & = & 15 - 1 \end{array} \end{aligned}

xx) =x-1f(15) =15-1

Then, perform the subtraction on the right side of the function rule.
::然后在函数规则右侧执行减法。

\begin{aligned} \begin{array}{rcl} f (15) & = & 15 - 1 \\ f (15) & = & 14 \end{array} \end{aligned}

::f( 15) =15 - 1f( 15) =14

The answer is 14.
::答案是14岁

If $\begin{aligned} f (x) = x - 1 \end{aligned}$ find $\begin{aligned} f (17) \end{aligned}$ .
::如果f(x) =x-1 发现f(17) 。

Next, substitute 17 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::接下来,在函数规则中,以17取代`x ' 。

\begin{aligned} \begin{array}{rcl} f (x) & = & x - 1 \\ f (17) & = & 17 - 1 \end{array} \end{aligned}

xx) =x-1f(17) =17-1

Then, perform the subtraction on the right side of the function rule.
::然后在函数规则右侧执行减法。

\begin{aligned} \begin{array}{rcl} f (17) & = & 17 - 1 \\ f (17) & = & 16 \end{array} \end{aligned}

::f( 17) =17- 1f( 17) =16

The answer is 16.
::答案是16岁

All of the $\begin{aligned} f (x) \end{aligned}$ values calculated using the function rule match the $\begin{aligned} f (x) \end{aligned}$ values given in the table.
::使用函数规则计算的所有 f(x) 值与表中给出的 f(x) 值相符。

The function rule $\begin{aligned} f (x) = x - 1 \end{aligned}$ represents the information given in the table.
::函数规则f(x)=x-1代表表格中提供的信息。

Example 3
::例3

Write a function rule using function notation to represent the information given in the following table.
::写入函数规则, 使用函数符号表示下表提供的信息。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} f (x) \end{aligned}$
12	6
9	4.5
7	3.5
4	2

First, look at the two columns of given numbers to determine a pattern between the values.
::首先,看看给定数字的两列,以确定数值之间的模式。

The numbers in the second column are one-half the numbers in the first column.
::第二列的数字是第一列数字的一半。

Next, write a function rule, using function notation, to represent the pattern.
::下一步,写入函数规则,使用函数符号表示模式。

\begin{aligned} f (x) = \frac{1}{2} x \end{aligned}

xx)=12x

Next, test the function rule for each of the given ‘ $\begin{aligned} x \end{aligned}$ ’ values.
::下一步,测试给定的每个 " x " 值的函数规则。

If $\begin{aligned} f (x) = \frac{1}{2} x \end{aligned}$ find $\begin{aligned} f (12) \end{aligned}$ .
::如果 f( x) =12x 找到 f( 12) 。

Next, substitute 12 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::接下来,在函数规则中以12取代 " x " 。

\begin{aligned} \begin{array}{rcl} f (x) & = & \frac{1}{2} x \\ f (12) & = & \frac{1}{2} (12) \end{array} \end{aligned}

xx)=12xf(12)=12(12)

Then, perform the multiplication on the right side of the function rule.
::然后在函数规则右侧执行乘法。

\begin{aligned} \begin{array}{rcl} f (12) & = & \frac{1}{2} (12) \\ f (12) & = & 6 \end{array} \end{aligned}

::f( 12) = 12( 12) f( 12) = 6

The answer is 6.
::答案是6个

If $\begin{aligned} f (x) = \frac{1}{2} x \end{aligned}$ find $\begin{aligned} f (9) \end{aligned}$ .
::如果f(x)=12x发现f(9) 。

Next, substitute 9 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::接下来,在函数规则中,将 " x " 改为 " 9 " 。

\begin{aligned} \begin{array}{rcl} f (x) & = & \frac{1}{2} x \\ f (9) & = & \frac{1}{2} (9) \end{array} \end{aligned}

::f(x) = 12xf(9) = 12(9)

Then, perform the multiplication on the right side of the function rule.
::然后在函数规则右侧执行乘法。

\begin{aligned} \begin{array}{rcl} f (9) & = & \frac{1}{2} (9) \\ f (9) & = & 4.5 \end{array} \end{aligned}

::f( 9) = 12( 9) f( 9) = 4. 5

The answer is 4.5.
::答案是4.5。

If $\begin{aligned} f (x) = \frac{1}{2} x \end{aligned}$ find $\begin{aligned} f (7) \end{aligned}$ .
::如果 f(x) = 12x 找到 f(7) 。

Next, substitute 7 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::接下来,在函数规则中将 " x " 改为 " 7 " 。

\begin{aligned} \begin{array}{rcl} f (x) & = & \frac{1}{2} x \\ f (7) & = & \frac{1}{2} (7) \end{array} \end{aligned}

xx)=12xf(7)=12(7)

Then, perform the multiplication on the right side of the function rule.
::然后在函数规则右侧执行乘法。

\begin{aligned} \begin{array}{rcl} f (7) & = & \frac{1}{2} (7) \\ f (7) & = & 3.5 \end{array} \end{aligned}

::f(7)=12(7)-f(7)=3.5

The answer is 3.5.
::答案是3.5

If $\begin{aligned} f (x) = \frac{1}{2} x \end{aligned}$ find $\begin{aligned} f (4) \end{aligned}$ .
::如果 f( x) =12x 找到 f(4) 。

Next, substitute 4 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::接下来,在函数规则中,将 " x " 替换为 " x " 。

\begin{aligned} \begin{array}{rcl} f (x) & = & \frac{1}{2} x \\ f (4) & = & \frac{1}{2} (4) \end{array} \end{aligned}

xx)=12xf(4)=12(4)

Then, perform the multiplication on the right side of the function rule.
::然后在函数规则右侧执行乘法。

\begin{aligned} \begin{array}{rcl} f (4) & = & \frac{1}{2} (4) \\ f (4) & = & 2 \end{array} \end{aligned}

::f(4)=12(4)(4)(4)=2

The answer is 2.
::答案是2。

The function rule $\begin{aligned} f (x) = \frac{1}{2} x \end{aligned}$ represents the information given in the table.
::函数规则f(x)=12x代表表格中提供的信息。

Example 4
::例4

Write a function rule using function notation to represent the information given in the following table.
::写入函数规则, 使用函数符号表示下表提供的信息。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} f (x) \end{aligned}$
9	-27
11	-33
15	-45
16	-48

First, look at the two columns of given numbers to determine a pattern between the values.
::首先,看看给定数字的两列,以确定数值之间的模式。

The numbers in the second column are negative three times the numbers in the first column.
::第二列的数字为负数,是第一列数字的三倍。

Next, write a function rule, using function notation, to represent the pattern.
::下一步,写入函数规则,使用函数符号表示模式。

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

fx) 3x

Next, test the function rule for each of the given ‘ $\begin{aligned} x \end{aligned}$ ’ values.
::下一步,测试给定的每个 " x " 值的函数规则。

If $\begin{aligned} f (x) = - 3 x \end{aligned}$ find $\begin{aligned} f (9) \end{aligned}$ .
::如果f(x)%%3x找到f(9) 。

Next, substitute 9 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::接下来,在函数规则中,将 " x " 改为 " 9 " 。

\begin{aligned} \begin{array}{rcl} f (x) & = & - 3 x \\ f (9) & = & - 3 (9) \end{array} \end{aligned}

xx) 3xf(9) 3x(9)

Then, perform the multiplication on the right side of the function rule.
::然后在函数规则右侧执行乘法。

\begin{aligned} \begin{array}{rcl} f (9) & = & - 3 (9) \\ f (9) & = & - 27 \end{array} \end{aligned}

::f(9)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

The answer is -27.
::答案是 -27

If $\begin{aligned} f (x) = - 3 x \end{aligned}$ find $\begin{aligned} f (11) \end{aligned}$ .
::如果f(x)%%%3x找到f( 11) 。

Next, substitute 11 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::接下来,在函数规则中,以 " x " 代替11。

\begin{aligned} \begin{array}{rcl} f (x) & = & - 3 x \\ f (11) & = & - 3 (11) \end{array} \end{aligned}

xx) 3xf(11) 3x(11)

Then, perform the multiplication on the right side of the function rule.
::然后在函数规则右侧执行乘法。

\begin{aligned} \begin{array}{rcl} f (11) & = & - 3 (11) \\ f (11) & = & - 33 \end{array} \end{aligned}

::f( 11)... 3( 11) f( 11)... 33

The answer is -33.
::答案是 -33

If $\begin{aligned} f (x) = - 3 x \end{aligned}$ find $\begin{aligned} f (15) \end{aligned}$ .
::如果f(x)%%3x发现f(15) 。

Next, substitute 15 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::接下来,在函数规则中,以15取代 " x " 。

\begin{aligned} \begin{array}{rcl} f (x) & = & - 3 x \\ f (15) & = & - 3 (15) \end{array} \end{aligned}

xx) 3xf(15) 3x(15)

Then, perform the multiplication on the right side of the function rule.
::然后在函数规则右侧执行乘法。

\begin{aligned} \begin{array}{rcl} f (15) & = & - 3 (15) \\ f (15) & = & - 45 \end{array} \end{aligned}

::f(15)___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

The answer is -45.
::答案是 -45

If $\begin{aligned} f (x) = - 3 x \end{aligned}$ find $\begin{aligned} f (16) \end{aligned}$ .
::如果f(x)%%%3x发现f(16) 。

Next, substitute 16 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::其次,在函数规则中,将 " x " 改为 " x " 。

\begin{aligned} \begin{array}{rcl} f (x) & = & - 3 x \\ f (16) & = & - 3 (16) \end{array} \end{aligned}

xx) 3xf(16) 3x(16)

Then, perform the multiplication on the right side of the function rule.
::然后在函数规则右侧执行乘法。

\begin{aligned} \begin{array}{rcl} f (16) & = & - 3 (16) \\ f (16) & = & - 48 \end{array} \end{aligned}

::f(16)_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

The answer is -48.
::答案是 -48

The function rule $\begin{aligned} f (x) = - 3 x \end{aligned}$ represents the information given in the table.
::函数规则f(x)%3x代表表格中提供的信息。

Example 5
::例5

$\begin{aligned} x \end{aligned}$	$\begin{aligned} f (x) \end{aligned}$
1	4
2	7
3	10
4	13

First, look at the two columns of given numbers to determine a pattern between the values.
::首先,看看给定数字的两列,以确定数值之间的模式。

The numbers in the second column are one greater than three times the values in the first column.
::第二列的数字是第一列数值的三倍以上。

Next, write a function rule, using function notation, to represent the pattern.
::下一步,写入函数规则,使用函数符号表示模式。

\begin{aligned} f (x) = 3 x + 1 \end{aligned}

xx)=3x+1

Next, test the function rule for each of the given ‘ $\begin{aligned} x \end{aligned}$ ’ values.
::下一步,测试给定的每个 " x " 值的函数规则。

If $\begin{aligned} f (x) = 3 x + 1 \end{aligned}$ find $\begin{aligned} f (1) \end{aligned}$ .
::如果 f(x) = 3x+1 找到 f(1) 。

Next, substitute 1 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::其次,在函数规则中,将 " x " 改为 " x " 。

\begin{aligned} \begin{array}{rcl} f (x) & = & 3 x + 1 \\ f (1) & = & 3 (1) + 1 \end{array} \end{aligned}

::f(x)=3x+1f(1)=3(1)+1

Then, perform the multiplication on the right side of the function rule.
::然后在函数规则右侧执行乘法。

\begin{aligned} \begin{array}{rcl} f (1) & = & 3 (1) + 1 \\ f (1) & = & 4 \end{array} \end{aligned}

::f(1)=3(1)+1f(1)=4

The answer is 4.
::答案是4。

If $\begin{aligned} f (x) = 3 x + 1 \end{aligned}$ find $\begin{aligned} f (2) \end{aligned}$ .
::如果 f(x) = 3x+1 找到 f(2) 。

Next, substitute 2 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::其次,在函数规则中,将 " x " 改为 " 2 " 。

\begin{aligned} \begin{array}{rcl} f (x) & = & 3 x + 1 \\ f (2) & = & 3 (2) + 1 \end{array} \end{aligned}

::f(x)=3x+1f(2)=3(2)+1

Then, perform the multiplication on the right side of the function rule.
::然后在函数规则右侧执行乘法。

\begin{aligned} \begin{array}{rcl} f (2) & = & 3 (2) + 1 \\ f (2) & = & 7 \end{array} \end{aligned}

::f(2)=3(2)+1f(2)=7

The answer is 7.
::答案是7个

If $\begin{aligned} f (x) = 3 x + 1 \end{aligned}$ find $\begin{aligned} f (3) \end{aligned}$ .
::如果 f(x) = 3x+1 找到 f(3) 。

Next, substitute 3 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::接下来,在函数规则中,将 " x " 改为 " 3 " 。

\begin{aligned} \begin{array}{rcl} f (x) & = & 3 x + 1 \\ f (3) & = & 3 (3) + 1 \end{array} \end{aligned}

::f(x)=3x+1f(3)=3(3)+1

Then, perform the multiplication on the right side of the function rule.
::然后在函数规则右侧执行乘法。

\begin{aligned} \begin{array}{rcl} f (3) & = & 3 (3) + 1 \\ f (3) & = & 10 \end{array} \end{aligned}

::f(3)=3(3)+1f(3)=10

The answer is 10.
::答案是10个

If $\begin{aligned} f (x) = 3 x + 1 \end{aligned}$ find $\begin{aligned} f (4) \end{aligned}$ .
::如果 f(x) =3x+1 找到 f(4) 。

Next, substitute 4 for ‘ $\begin{aligned} x \end{aligned}$ ’ in the function rule.
::接下来,在函数规则中,将 " x " 替换为 " x " 。

\begin{aligned} \begin{array}{rcl} f (x) & = & 3 x + 1 \\ f (4) & = & 3 (4) + 1 \end{array} \end{aligned}

::f(x)=3x+1f(4)=3(4)+1

Then, perform the multiplication on the right side of the function rule.
::然后在函数规则右侧执行乘法。

\begin{aligned} \begin{array}{rcl} f (4) & = & 3 (4) + 1 \\ f (4) & = & 13 \end{array} \end{aligned}

::f(4)=3(4)+1f(4)=13

The answer is 13.
::答案是13岁

The function rule $\begin{aligned} f (x) = 3 x + 1 \end{aligned}$ represents the information given in the table.
::函数f(x)=3x+1代表表格中提供的信息。

Review
::回顾

1. Write a function rule for the following data.
::1. 为下列数据写出函数规则。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} f (x) \end{aligned}$
9	11
11	9
15	5
16	4

2. Write a function rule for the following data.
::2. 为下列数据写出函数规则。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} f (x) \end{aligned}$
-2	-6
0	-4
2	-2
4	0

3. Write a function rule for the following data.
::3. 为下列数据写出函数规则。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} f (x) \end{aligned}$
0	0
1	2
4	8
5	10

4. Write a function rule for the following data.
::4. 为下列数据写出函数规则。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} f (x) \end{aligned}$
1	0
2	2
4	6
8	14

5. Write a function rule for the following data.
::5. 为下列数据写出函数规则。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} f (x) \end{aligned}$
2	1
4	2
8	4
10	5
18	9

6. Write a function rule for the following data.
::6. 为下列数据写出函数规则。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} f (x) \end{aligned}$
6	2
9	3
15	5
21	7
30	10

7. Write a function rule for the following data.
::7. 为下列数据写出函数规则。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} f (x) \end{aligned}$
2	3
9	10
15	16
21	22
30	31

8. Write a function rule for the following data.
::8. 为下列数据写出函数规则。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} f (x) \end{aligned}$
3	-6
9	-18
15	-30
20	-40
24	-48

Solve the following problem.
::解决以下问题。

9. Sandwiches cost $3.45 each. Write a function rule for the cost, $\begin{aligned} c \end{aligned}$ , for a number of sandwiches, $\begin{aligned} s \end{aligned}$ .
::9. 三明治每人费用为3.45美元。

10. Now, find the cost of 3 sandwiches.
::10. 现在,找出3个三明治的费用。

11. Find the cost of 6 sandwiches.
::11. 找出6个三明治的费用。

12. Find the cost of 9 sandwiches.
::12. 找出9个三明治的费用。

13. Find the cost of 2 sandwiches.
::13. 找出2个三明治的费用。

14. Find the cost of 8 sandwiches.
::14. 寻找8个三明治的费用。

15. Find the cost of a dozen sandwiches.
::15. 找出一打三明治的费用。

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

Function Rule ::职能规则

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)

Resources ::资源

Function Rule
::职能规则

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)

Resources
::资源