Section: 函数符号 | 5.5 函数符号

Section outline

You model the costs and the revenue for a business. On Friday afternoon, you come up with a model for the costs, $\begin{aligned} y = 90 x + 1800 \end{aligned}$ , and the revenue, $\begin{aligned} y = 120 x \end{aligned}$ . Then, the workday ends and you go hang out with some friends. On Saturday, you have a full day and end the night partying. Sunday is another busy day having fun with your friends and family. On Monday, you come back to work and you have two equations on your desk. You try to remember what you did on Friday. Which of these equations is which? Wouldn't it be easier if the equations had names? That is exactly what function notation is—a way to name functions.
::商业成本和收入的模型。星期五下午, 你拿出一个成本的模型, y=90x+1800, 和收入的模型, y=120x。然后, 工作日结束, 你和一些朋友一起出去玩。星期六, 你有一个一整天的聚会, 晚上结束。星期天是另一个忙碌的一天, 和你的朋友和家人一起玩。星期一, 你回来工作, 桌上有两个方程式。你尽量记住你周五做了些什么。这些方程式中的哪一个是哪一个? 如果公式有名字的话, 会不会更容易? 这就是符号的作用—— 一种命名功能的方式。

Function Notation
::函数符号

We do not use y equals as an expression for most practical applications. That is largely a construct used in algebra courses. In most practical applications, we use function notation, which allows us to indicate a name for the function . It is also a shorter way of expressing the solutions to an equation .
::我们不使用 Y 等同的表达方式来表达最实际的应用。这基本上是代数课程中的一种构造。在大多数实际应用中,我们使用函数符号,这使我们能够为函数指定一个名称。它也是表达方程式解决方案的较短方式。

Say we want to write the function, $\begin{aligned} y = 200 x + 3 \end{aligned}$ , in function notation. We will replace the y with f(x) , f for function. This is read "f of x." So, our equation now looks like $\begin{aligned} f (x) = 200 x + 3 \end{aligned}$ . The parts are: x is the input and we put that in the " data-term="Parentheses" role="term" tabindex="0"> parentheses , $\begin{aligned} 200 x + 3 \end{aligned}$ is the rule we will follow, and f(x) is the output.
::说我们要在函数符号中写入函数, y=200x+3 。我们将用 f( x), f 替换 y 。这读为“ f of x ” 。因此, 我们的方程式现在看起来像 f( x) = 200x+3 。部件是: x 是输入, 我们把它放在括号中, 200x+3 是我们要遵循的规则, f( x) 是输出。

by CK-12 explains function notation.
::在 CK-12 中解释函数符号。

Let's consider writing some other equations with function notation.
::让我们考虑用函数符号来写一些其他方程。

Example 1
::例1

Write the cost equation, $\begin{aligned} y = 90 x + 1800 \end{aligned}$ , and the revenue equation, $\begin{aligned} y = 120 x \end{aligned}$ , with function notation.
::写成本方程式,y=90x+1800, 收入方程式,y=120x, 带有函数符号。

Solution: Since the first equation represents the costs, let's call it C(x). (Read "C of x.") So, $\begin{aligned} C (x) = 90 x + 1800 \end{aligned}$ . The second equation is the revenue, so we will call it R(x) (read "R of x"). Thus, $\begin{aligned} R (x) = 120 x \end{aligned}$ .
::解答: 由于第一个方程式代表成本, 我们称之为 C( x) 。 (读“ C of x. ” ) 。所以, C( x) = 90x+1800。第二个方程式是收入, 所以我们称之为 R( x) (读“ R of x” ) 。因此, R( x) =120x 。

WARNING
::警告

Unfortunately, the notation looks like multiplication . Do not multiply by x .
::不幸的是,符号看起来像乘法。不要乘以 x 。

For example, $\begin{aligned} f (x) = 3 x + 2 \end{aligned}$ is not the same as $\begin{aligned} (3 x + 2) \cdot x = 3 x^{2} + 2 x \end{aligned}$ .
::例如,f(x)=3x+2与(3x+2)x=3x2+2x不相同。

Example 2
::例2

Evaluate $\begin{aligned} f (x) = 4 x - 7 \end{aligned}$ for $\begin{aligned} x = 3 \end{aligned}$ and $\begin{aligned} x = - 2 \end{aligned}$ .
::x=3 和 x2 的 f(x) = 4x-7 评价 f(x) = 4x- 7。

Solution: We are given two inputs or x -values. First, we put each input in the parentheses. Then, we replace x in the rule with each value. We will follow the rule that says to multiply by 4 and then to subtract 7. This will give us our output.
::解答: 我们有两个输入值或 x 值。首先, 我们把每个输入值都放在括号中。然后, 我们用每个值替换规则中的 x 。我们将遵循规则, 要求乘以 4 , 然后减去 7 。这将给我们输出。

$\begin{aligned} f (3) = 4 (3) - 7 = 12 - 7 = 5 & f (3) = 5 \\ f (- 2) = 4 (- 2) - 7 = - 8 - 7 = - 15 & f (- 2) = - 15 \end{aligned}$

::f(3)=4(3)-7=12-7=5f(3)=5f(-2)=4(-2)=4(2)-7(8)-7(5)(-2)

Since f(x) indicates the y -value, these calculations indicate two points on the graph of $\begin{aligned} f (x) = 4 x - 7 \end{aligned}$ : (3,5) and (-2,-15).
::由于f(x)表示y值,这些计算表明f(x)=4x-73,5)和(-2,15)图中的两点。

by CK-12 demonstrates how to evaluate functions given in function notation.
::CK-12 显示如何评价函数符号中赋予的职能。

Example 3
::例3

Evaluate $\begin{aligned} g (x) = \sqrt{x} + 1 \end{aligned}$ for $\begin{aligned} x = 25, x = \frac{1}{4}, x = a, and x = a + b \end{aligned}$ .
::x=25,x=14,x=a,x=a,和x=a+b,评价 g(x)=x+1。

Solution: Just as we did in Example 2, we will put each of these values or variables in the parentheses and then follow the rule that says take the square root of the input and then add 1.
::解答:和例2一样,我们将将这些数值或变量中的每一个置于括号中,然后遵循规则,即从输入的平方根中取出输入的平方根,然后添加1。

$\begin{aligned} g (25) = \sqrt{25} + 1 = 5 + 1 = 6 & g (25) = 6 \\ g (\frac{1}{4}) = \sqrt{\frac{1}{4}} + 1 = \frac{\sqrt{1}}{\sqrt{4}} + 1 = \frac{1}{2} + 1 = \frac{3}{2} & g (\frac{1}{4}) = \frac{3}{2} \\ g (a) = \sqrt{a} + 1 \\ g (a + b) = \sqrt{a + b} + 1 \end{aligned}$

::g( 25) =25+1=5+1=5+1=6g( 25) =6g( 14) =14+1=14+1=14+1=12+1=32g( 14) =32g( a) =a+1g( a+b) =a+b+1

WARNING
::警告

$\begin{aligned} f (a + b) \neq f (a) + f (b) \\ f (a - b) \neq f (a) - f (b) \\ f (a b) \neq f (a) \cdot f (b) \\ f (a b) \neq a f (b) \neq f (a) b \end{aligned}$

Example 4
::例4

The height of a projectile launched into the air can be modeled by $\begin{aligned} h (t) = t^{2} - 4 t + 12 \end{aligned}$ . Find the height after 5 seconds.
::射入空气的射弹高度可以通过 h(t) =t2 - 4t+12进行模拟。在5秒后查找高度。

Solution: Notice that x does not have to be the input variable . We often use t to represent time. To find the height of the projectile after 5 seconds, we need to evaluate the function when $\begin{aligned} t = 5 \end{aligned}$ .
::解析 : 注意 x 不必是输入变量。我们经常使用 t 来代表时间。要在 5 秒后找到投射体的高度, 我们需要在 t= 5 时评估函数。

$\begin{aligned} h (5) = 5^{2} - 4 (5) + 12 = 25 - 20 + 12 = 17 \end{aligned}$

::h(5)=52-4(5)+12=25-20+12=17

After 5 seconds, the projectile is 17 feet in the air.
::5秒后,弹体在空中17英尺处

Example 5
::例5

For what value(s) of x is $\begin{aligned} f (x) = 3 x - 2 \end{aligned}$ equal to 10?
::x 的值是 F( x) =3x-2 等于 10?

Solution: Here we are given the y -value and asked to find x . We replace f(x) with 10 and solve.
::解答:在这里,我们得到了Y值,并要求找到 x。我们用 10 替换 f(x),然后解答。

$\begin{aligned} 10 = 3 x - 2 \\ \underline{+ 2 + 2} \\ \frac{12}{3} = \frac{3 x}{3} \\ 4 = x \end{aligned}$
Therefore , $\begin{aligned} f (4) = 10 \end{aligned}$ .
::10=3x-2+2+2 +2_123=3x34=xxf(4)=10。

by CK-12 demonstrates how to find inputs given outputs.
::CK-12展示了如何找到特定产出的投入。

Piecewise-defined Functions
::小数定义函数

In the section on domain and range , we saw some graphs of piecewise-defined functions. Here we take a look at how they are formed.
::在关于域和范围的一节中,我们看到了一些图表,其中显示了片段定义的函数。在这里,我们来看看这些函数是如何形成的。

A piecewise-defined function is a function that is defined as different pieces on different intervals. They look like this
::papwikefisled 函数是按不同间隔被定义为不同块的函数。它们看起来像这样

$\begin{aligned} f (x) = {\begin{cases} Rule 1 & Interval 1 \\ Rule 2 & Interval 2 \\ Rule 3 & Interval 3 \\ . & . \\ . & . \\ . & . \end{cases} \end{aligned}$

::f(x) 规则1 规则2 规则2 规则3

Note the dots indicate there can be an infinite number of pieces and rules can be defined at a single point as well.
::请注意,点表示可以有无限数量的项目,规则也可以在一个点上界定。

To evaluate a piecewise-defined function, first you find the interval or point your input value satisfies. Then, you follow the rule associated with those x- values.
::要评价一个按片段定义的函数,首先,您会发现输入值的间隔或点满足。然后,您会遵循与这些 X 值相关的规则。

Example 6
::例6

Evaluate $\begin{aligned} f (x) = {\begin{cases} - x + 1 & x < 0 \\ 4 & x = 0 \\ | x | & x > 0 \end{cases} \end{aligned}$ for $\begin{aligned} x = - 2, x = 0, x = 2 \end{aligned}$ .
::以 f( x)\\ xxxxxx x+1x < 04x=0\\ xxx>0 来评价 f( x) =xxx+1x < 04x=0xxxxxxx>0, x=0x=2。

Solution: Let us first consider $\begin{aligned} x = - 2 \end{aligned}$ , which satisfies the first interval. So, we follow the first rule: $\begin{aligned} f (- 2) = - (- 2) + 1 = 2 + 1 = 3 \end{aligned}$ .
::解答: 让我们首先考虑 x2, 它满足了第一个间隔。因此, 我们遵循第一条规则: f(-2) (-2)+1=2+1=3。

Zero is equal to itself, so we follow the second rule that says to output 4, that is, $\begin{aligned} f (0) = 4 \end{aligned}$ .
::零等于本身,所以我们遵循第二项规则即产出4,即f(0)=4。

Lastly, $\begin{aligned} x = 2 \end{aligned}$ is in the last interval, so we follow the last rule: $\begin{aligned} f (2) = | 2 | = 2. \end{aligned}$
::最后, x=2 是最后的间隔, 所以我们遵循了最后的规则 : f(2) 2 2 。

by CK-12 explains how evaluate piecewise-defined functions .
::使用 CK-12 来解释如何对按片段定义的函数进行评价。

How To Evaluate a Function and Auto-Generate a Table of Values With Desmos
::如何评价函数和自动生成带有脱mos的数值表

1. Input the function as you would if you were graphing.
::1. 在图形化时,按您的意愿输入函数。

Use $\begin{aligned} f (x) \end{aligned}$ instead of y .
::用f(x)代替y。

2 . Now, to evaluate the function, type $\begin{aligned} f (c) \end{aligned}$ to evaluate the function at c .
::2. 现在,为了评价函数,f(c)类类型用于评价 c.的函数。

3. To generate a table of values:
::3. 生成一个数值表:

Choose table from the Add Item tab in the upper-left corner. Change the variables at the top of the table to x and $\begin{aligned} f (x) \end{aligned}$ . Enter the values for x that you want to see in your table.
::从左上角的添加项选项卡中选择表格。将表格顶部的变量更改为 x 和 f(x)。输入您想要在表格中看到的 x 的值。

How to Evaluate a Function and Auto-Generate a Table of Values on a TI-83/84
::如何评价一个功能和自动生成TI-83/84的数值表

1. Input the function as you would if you were graphing.
::1. 在图形化时,按您的意愿输入函数。

Use $\begin{aligned} f (x) \end{aligned}$ instead of y .
::用f(x)代替y。

Next, on a TI-83 or TI-84, choose VARS → Y-VARS →Function→Choose the $\begin{aligned} Y_{n} \end{aligned}$ that you used in step 1.
::接下来,在TI-83或TI-84上, 选择您在第1步中使用的YY-VARS-FUNCY选择YN。

2 . Now, to evaluate the function, type $\begin{aligned} Y (c) \end{aligned}$ to evaluate the function at c .
::2. 现在,为了评价函数,Y(c)类型用于评价 c.的函数。

3. To generate a table of values:
::3. 生成一个数值表:

In Desmos, choose table from the Add Item tab in the upper-left corner. Change the variables at the top of the table to x and $\begin{aligned} f (x) \end{aligned}$ . Enter the values for x that you want to see in your table.
::在 Desmos 中,从左上角的添加项目选项卡中选择表格。将表格上方的变量更改为 x 和 f(x) 。输入您想要在表格中看到的 x 的值。

On a TI-83 or TI-84, choose TABLE . You may have to change your table settings to get the values you want to see generated in your table.
::在 TI-83 或 TI-84 上,请选择 Table 。您可能需要更改表格设置以获取您想要在表格中看到的数值。

Features: Splines: The Truth Behind Video Games
::特色:Splines:视频游戏背后的真相

A spline is a piecewise-defined function with polynomial function pieces that is smooth, that is, no corners or cusps are in the graph. The pieces connect at points called knots. Splines are often used in computer graphics to model the real-world settings on the screen. They are popular because of their ease of use and accuracy of evaluation, and their ability to approximate complex shapes through curve fitting. Because of these good properties, s plines are also often used in designing hulls or bodies for ships, airplanes and automobiles. ¹
::样条是一个小字形定义的函数, 带有光滑的多面函数块, 也就是说, 在图形中没有角或角。连接点在称为结节的点上。在计算机图形中, 常使用样条来模拟屏幕上的真实世界设置。它们很受欢迎, 因为它们使用方便, 评估准确, 并且能够通过曲线安装来接近复杂的形状。由于这些良好的特性, 样条也经常用于船舶、飞机和汽车的船体或船体设计中。

Cubic splines are particularly good for curve fitting and curve design. For example, the Irwin-Hall Distribution is often used in practical applications and probability. ²
::三次曲线样条对曲线的安装和曲线设计特别有用,例如,Irwin-Hall分布法经常用于实际应用和概率。

$\begin{aligned} f (x) = {\begin{cases} \frac{1}{4} (x + 2)^{3} & - 2 \leq x \leq - 1 \\ \frac{1}{4} (3 | x |^{3} - 6 x^{2} + 4) & - 1 \leq x \leq 1 \\ \frac{1}{4} (2 - x)^{3} & 1 \leq x \leq 2 \end{cases} \end{aligned}$

:xx) 14(x+2) 3- 2x) 114(3x) 3- 6x2+4) - 1x 114(2-x) 3- 3

by Christoph von Tycowicz shows the types of movements that can be modeled with splines.
::Christoph von Tycowicz展示了可以用样条模拟的运动类型。

Summary
::摘要

To evaluate a function, replace the variable in the rule with the expression inside the parentheses.
::要评价函数,将规则中的变量替换为括号内的表达式。

Review
::回顾

Find the given value of the function.
::查找函数的给定值。

1. $\begin{aligned} f (x) = \frac{1}{2} x - 5; f (8) \end{aligned}$
::1. f(x)=12x-5;f(8)

2. $\begin{aligned} g (x) = | x - 4 |; g (7) \end{aligned}$
::2. g(x) x-4;g(7)

3. $\begin{aligned} s (t) = 55 t + 200; s (3) \end{aligned}$
::3. s(t)=55t+200;s(3)

4. $\begin{aligned} R (p) = \frac{p - 9}{p + 6}; R (10) \end{aligned}$
::4. R(p) =p- 9p+6;R(10)

5. $\begin{aligned} Q (a) = a^{3} + 2 a^{2} + 5; Q (- 4) \end{aligned}$
::5. Q(a)=a3+2a2+5;Q(-4)

6. $\begin{aligned} f (x) = {\begin{cases} 1 & x < 1 \\ 3 x & x > 4 \end{cases}; f (5) and f (0) \end{aligned}$
::6. f(x)%1x<13xx>4;f(5)和f(0)

7. $\begin{aligned} g (x) = {\begin{cases} - x & x < 0 \\ x & x \geq 0 \end{cases}; g (- 3) and g (0) \end{aligned}$
::7. g(x)xxxx < 0xXX0;g(-3)和g(0)

For what values of x is $\begin{aligned} f (x) = 0 \end{aligned}$ ?
::x 的值是 f( x)=0 ?

8. $\begin{aligned} f (x) = - \frac{1}{4} x + 3 \end{aligned}$
::8. f(x)14x+3

9. $\begin{aligned} f (x) = | x - 1 | - 2 \end{aligned}$
::9. f(x) ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

10. $\begin{aligned} f (x) = 5 \end{aligned}$
::10.f(x)=5

Explore More
::探索更多

1. One common application of function notation is finding the slope of a secant line through a function. A secant line is a line that intersects the graph in two points.
::1. 函数标记的一个常见应用是,通过函数找到松散线的斜坡。

This approximates how the graph is changing over an interval. We call this the average rate of change. The formula is
::这大致相当于图形在一个间隔内如何变化。我们称此为平均变化率。公式是

$\begin{aligned} A R C = \frac{f (x_{2}) - f (x_{1})}{x_{2} - x_{1}} \end{aligned}$

::ARC = f( x2) - f( x1) x2 - x1

Notice this is the change in the y -values over the change in x -values, so it is a slope.
::注意这是 x 值变化之后 Y 值的变化, 所以它是一个斜坡。

Find the ARC for $\begin{aligned} f (x) = x^{2} + 1 \end{aligned}$ when $\begin{aligned} x_{1} = - 1 \end{aligned}$ and $\begin{aligned} x_{2} = 3 \end{aligned}$ .
::查找 f( x) =x2+1 的 ARC, 当 x1\\\\\ 1 和 x2= 3 时查找 ARC 。

2. Suppose you worked at an animal rescue caring for the dogs, and you wanted to determine the age of each of the dogs in "dog years," based on the age in human years. The following piecewise-defined function can be used to convert the years that a dog has been alive to the dog's age in "dog years." In this function t represents the whole number years that the dog has been alive and D(t) represents the dog's age in "dog years."
::2. 假设你在动物营救中照顾狗,你想根据人年年龄来决定每只狗在“狗年”中的年龄,以下的片段定义功能可用于将狗活到的年数转换为狗年的年数。在这个函数中,t 表示狗活了多少年,D(t) 表示狗在“狗年”中的年数。

$\begin{aligned} D (t) = {\begin{cases} 10.5 t & 0 \leq t \leq 2 \\ 4 t + 21 & t > 2 \end{cases} \end{aligned}$

::D(t) 10.5t00t 24t+21t>2

Find the age of a dog in dog years if it is 5 years old.
::如果狗狗是5岁,在狗年中找到狗的年龄。

3. Suppose the value V of a digital camera t years after it was bought is represented by the function $\begin{aligned} V (t) = 875 - 50 t \end{aligned}$ . Can you determine the value of V(4) and explain what the solution means in the context of this problem? Can you determine the value of t when V(t)=525 and explain what this situation represents? What was the original cost of the digital camera?
::3. 假设数字相机购买后两年的V值由函数V(t)=875-50t表示。您能否确定 V(4) 的价值,并解释在此问题上解决方案意味着什么?您能否确定V(t)=525 值,并解释这种情况代表什么?数字相机的原始成本是多少?

4. The emergency brake cable in a truck parked on a steep hill breaks and the truck rolls down the hill. The distance in feet, d, that the truck rolls is represented by the function $\begin{aligned} d (t) = 0.5 t^{2} \end{aligned}$ where $\begin{aligned} t \end{aligned}$ is in seconds. How far will the truck roll after 9 seconds? How long will it take the truck to hit a tree which is at the bottom of the hill 600 feet away? Round your answer to the nearest second.
::4. 一辆停在陡峭山丘间断处的卡车的紧急刹车电缆,卡车从山上滚下来,足足的距离,d,卡车卷号用函数d(t)=0.5t2代表,其位置几秒钟,卡车在9秒后滚动到多远?卡车要用多久才能撞到600英尺外山底的一棵树?

5. Suppose you just purchased a used car, and the number of miles on the odometer can be represented by the equation $\begin{aligned} y = x + 30, 000 \end{aligned}$ , where y is the number of miles on the odometer, and x is the number of miles you have driven it. Could you convert this equation to function notation? How many miles will be on the odometer if you drive the car 700 miles?
::5. 假设你刚刚购买了一辆二手车,而气压计上的英里数可以用公式y=x+30,000来表示,其中y是气压计上的英里数,x是驱动气压计的英里数,X是驱动气流的英里数。你能将这个方程转换成一个标记吗?如果驾驶700英里的车,气压计上将有多少英里?

Answers to Review and Explore More Problems

::对审查和探讨更多问题的答复

Please see the Appendix.

::请参看附录。

References
::参考参考资料

1. "Spline (mathematics)," last edited May 10, 2017, .
::2017年5月10日编辑。

2. "Irwin-Hall Distribution," last edited December 16, 2016, .
::2. “Irwin-Hall发行”, 2016年12月16日经编辑。

Section outline

Function Notation ::函数符号

Example 1 ::例1

WARNING ::警告

Example 2 ::例2

Example 3 ::例3

WARNING ::警告

Example 4 ::例4

Example 5 ::例5

Piecewise-defined Functions ::小数定义函数

Example 6 ::例6

How To Evaluate a Function and Auto-Generate a Table of Values With Desmos ::如何评价函数和自动生成带有脱mos的数值表

How to Evaluate a Function and Auto-Generate a Table of Values on a TI-83/84 ::如何评价一个功能和自动生成TI-83/84的数值表

Features: Splines: The Truth Behind Video Games ::特色:Splines:视频游戏背后的真相

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers to Review and Explore More Problems ::对审查和探讨更多问题的答复

References ::参考参考资料

Function Notation
::函数符号

Example 1
::例1

WARNING
::警告

Example 2
::例2

Example 3
::例3

WARNING
::警告

Example 4
::例4

Example 5
::例5

Piecewise-defined Functions
::小数定义函数

Example 6
::例6

How To Evaluate a Function and Auto-Generate a Table of Values With Desmos
::如何评价函数和自动生成带有脱mos的数值表

How to Evaluate a Function and Auto-Generate a Table of Values on a TI-83/84
::如何评价一个功能和自动生成TI-83/84的数值表

Features: Splines: The Truth Behind Video Games
::特色:Splines:视频游戏背后的真相

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers to Review and Explore More Problems

::对审查和探讨更多问题的答复

References
::参考参考资料