Course: 3.1 函数和逆函数

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函数和反函数

You may not realize it, but you have dealt with inverse functions for most of your life. Inverse functions are functions which 'undo' each other. For instance, $\begin{aligned} 3 t = T \end{aligned}$ is the function to convert teaspoons to tablespoons. The inverse: $\begin{aligned} t = \frac{T}{3} \end{aligned}$ converts from tablespoons back to teaspoons.
::您可能没有意识到它, 但您一生的大部分时间都处理过反函数。反函数是“ 互不” 的函数。例如, 3t=T 是将茶匙转换成茶匙的函数。反之 : t=T3 转换成茶匙。

Inverse functions are also used in geometry. Consider the following:
::几何中也使用逆函数。考虑以下各点:

You are asked to design the new box for the iMp3 player that your company makes. You know that the iMp3 is 5.4in tall, 2.3in wide, and .5in thick.
::您被要求为您公司制造的 iMp3 播放器设计新框。您知道 iMp3 高5. 4, 宽2.3, 厚5. 5 。

What is the volume of a box that will just fit the iMp3?
::一个符合iMp3的盒子的体积是多少?

If you knew that the iMp3 2 was soon to be released, and it had 10% less volume without changing height or width, could you prepare a box for it?
::如果你知道iMp3 2 很快即将发布, 且其体积少10%而不改变高度或宽度, 你能准备一个盒子吗?

Can you identify the inverse functions that might be applicable in this scenario?
::您能否确定可能适用于此情景的反函数 ?

Functions and Inverses
::函数和反函数

Consider the functions $\begin{aligned} t (x) = \frac{5}{9} (x - 32) \end{aligned}$ and $\begin{aligned} f (x) = \frac{9}{5} x + 32 \end{aligned}$ together. These two functions are inverses.
::将函数 t( x) = 59( x-32) 和 f( x) = 95x+32 一起考虑。这两个函数是反向的。

Informally, if two functions are inverses , then the input of one function is the output of the other.
::非正式地说,如果两个功能是反向的,那么一个函数的投入就是另一个函数的产出。

Formally, the inverse of a function is defined as follows:
::在形式上,一项职能的反面定义如下:

Functions f( x ) and g ( x ) are said to be inverses if f(g(x)) = g(f(x)) = x.
::如果f(g(x)=g(f(x))=x,函数f(x)和g(x)据说是反数。

Or, using the composite function notation: f ◦ g = g ◦ f = x.
::或者,使用复合函数符号:fg=gf=x。

The following notation is used to indicate inverse functions: If f ( x ) and g ( x ) are inverse functions, then f ( x ) = g ^-1 ( x ) and g ( x ) = f ^-1 ( x ) The following notation is also used: f = g ^-1 and g = f ^-1 . Note that f ^-1 (x) does not equal $\begin{aligned} \frac{1}{f (x)} \end{aligned}$ .
::以下标记用于表示反函数: Iff(x) 和 g(x) 是反函数, 然后f(x) =g-1(x) 和g(x) =f-1(x) 也使用以下标记: f=g-1andg=f-1。注意 F-1(x) 并不等于 1f(x) 。

Informally, we can identify the inverse of a function as the relation we obtain by switching the domain and range of the function. Because of this definition, you can find an inverse by switching the roles of x and y in an equation. For example, consider the function g ( x ) = 2 x . This is the line y = 2 x. If we switch x and y , we get the equation x = 2 y. Dividing both sides by 2, we get y = 1/2 x . Therefore the functions g(x) = 2 x and y = 1/2 x are inverses. Using function notation, we can write y = 1/2 x as g ^-1 (x) = 1/2 x .
::非正式地,我们可以辨别函数的反向, 即我们通过切换函数的域和范围而获得的关系。由于此定义, 您可以通过切换公式中的 x 和 y 的作用来找到反向。例如, 考虑函数 g( x) = 2x 。这是线y = 2x。如果切换 x 和 y, 我们得到方程式 x = 2y。将两侧切换为 2, 我们得到 y = 1/2 x 。因此, 函数 g( x) = 2x 和 y = 1/2 x 是反向的。使用函数符号, 我们可以写入 y = 1/2 x 作为 g-1 (x) = 1/2 x 。

We can also analyze two functions and determine whether or not they are inverses. Look carefully at the formal :
::我们还可以分析两种功能并确定它们是否反向。

Two functions f ( x ) and g ( x ) are inverses if and only if f(g(x)) = g(f(x)) = x .
::两个函数 f( x) 和 g( x) 是反向的, 只有if( g( x)) = g( f( x)) = x 。

That means that just as we can find the inverse of a function by identifying one that fits the definition, we can verify a possible inverse by testing it against the definition.
::这意味着,正如我们能够通过确定符合定义的函数,找到一种功能的反面,我们也可以根据定义测试一种可能的反向,从而核实一种可能的反向。

Examples
::实例

Example 1
::例1

Earlier, you were given a question about designing a box.
::早些时候,有人问你设计一个盒子的问题。

You were asked to design the new box for the iMp3 player that your company makes. You know that the iMp3 is 5.4in tall, 2.3in wide, and .5in thick.
::您被要求为您公司制作的 iMp3 播放器设计新框。您知道 iMp3 高5. 4, 宽2.3, 厚5. 5。

What is the volume of a box that will just fit the iMp3?
::一个符合iMp3的盒子的体积是多少?

Recall that the volume of a rectangular solid is given by: $\begin{aligned} V = l \cdot w \cdot h \end{aligned}$
::回顾矩形固体的体积是由: V=lwh给出的

$\begin{aligned} ∴ V = 6.21 i n^{3} \end{aligned}$
::V=6.21英寸3

If you knew that the iMp3 2 was soon to be released, and it had 10% less volume without changing height or width, could you prepare a box for it?
::如果你知道iMp3 2 很快即将发布, 且其体积少10%而不改变高度或宽度, 你能准备一个盒子吗?

The missing dimension for the new box is height: $\begin{aligned} .9 (6.21) = 5.4 \cdot 2.3 \cdot h \end{aligned}$
::新框缺少的尺寸是高度: 9(6. 21) = 5. 4. 3}h

$\begin{aligned} ∴ h = .45 i n \end{aligned}$
::h=45英寸

Can you identify the inverse functions that might be applicable in this scenario?
::您能否确定可能适用于此情景的反函数 ?

Two inverse functions used here are: %20%3D%20(l%20%5Ccdot%20w)%20%5Ccdot%20h"> $\begin{aligned} V (h) = (l \cdot w) \cdot h \end{aligned}$ and $\begin{aligned} h (v) = \frac{v}{l \cdot w} \end{aligned}$ .
::这里使用的两种反函数是: V =(lw) h 和 h(v) =vlw。

Verify that the two functions are inverses: Let $\begin{aligned} h = 2 \end{aligned}$ and assume a base area of $\begin{aligned} 6 \end{aligned}$ .
::验证这两个函数是反向的 : Let h=2 并假设基面积为 6 。

$\begin{aligned} V (2) = 6 \cdot 2 \end{aligned}$ ==> $\begin{aligned} V (2) = 12 \end{aligned}$
::V(2)=62 V(2)=12

$\begin{aligned} h (12) = 12 / 6 \end{aligned}$ ==> $\begin{aligned} h (12) = 2 \end{aligned}$
::h( 12) = 12/6 h( 12) = 2

This checks out, as $\begin{aligned} V (h (2)) = 2 = h (V (2)) \end{aligned}$ .
::检查结果为 V(h(2))=2=h(V(2))。

Example 2
::例2

In the United States, we measure temperature using the Fahrenheit scale. In other countries, people use the Celsius scale. The equation C = 5/9 ( F - 32) can be used to find C, the Celsius temperature, given F , the Fahrenheit temperature. If we write this equation using function notation, we have $\begin{aligned} t (x) = \frac{5}{9} (x - 32) \end{aligned}$ . The input of the function is a Fahrenheit temperature, and the output is a Celsius temperature. This function allows us to convert a Fahrenheit temperature into Celsius.
::在美国,我们用华氏级测量温度。在其他国家,人们使用摄氏级。方程式 C = 5/9 (F - 32) 可以用来查找 C, 摄氏温度, 给F, 给F, 法氏温度。如果我们用函数符号来写这个方程式, 我们的输入值是华氏温度, 输出值是摄氏度。这个函数允许我们将华氏温度转换为摄氏度。

Identify the inverse function of the equation above to get one that will convert Celsius to Fahrenheit.
::识别上面方程式的反函数, 以获得一个能将摄氏度转换为华氏值的方程式。

Fahrenheit to Celsius: $\begin{aligned} C = \frac{5}{9} (F - 32) \end{aligned}$
::华氏至摄氏度:C=59(F-32)

Start by isolating F:
::以孤立 F 开始 :

	$\begin{aligned} C = \frac{5}{9} (F - 32) \end{aligned}$
	$\begin{aligned} \frac{9}{5} C = \frac{9}{5} \times \frac{5}{9} (F - 32) \end{aligned}$
	$\begin{aligned} \frac{9}{5} C = F - 32 \end{aligned}$
	$\begin{aligned} \frac{9}{5} C + 32 = F \end{aligned}$

If we write this equation using function notation, we get $\begin{aligned} f (x) = \frac{9}{5} x + 32 \end{aligned}$ . For this function, the input is the Celsius temperature, and the output is the Fahrenheit temperature.
::如果我们使用函数符号写入此方程式, 我们就会得到 f( x) = 95x+32。对于此函数, 输入值为摄氏温度, 输出值为 Fahrenheit 温度。

Use the Celsius to Fahrenheit equation to convert 0 degrees Celsius into Fahrenheit.
::用摄氏度到华氏方程将0摄氏度转换成华氏度

If it is 0 degrees Celsius, then we have: $\begin{aligned} x = 0, f (0) = \frac{9}{5} (0) + 32 = 0 + 32 = 32 \end{aligned}$
::如果是0摄氏度,那么我们有:x=0,f(0)=95(0)+32=0+32=32

0 degrees Celsius = 32 degrees Fahrenheit
::0摄氏度=32华氏度

Example 3
::例3

Find the inverse of each function.
::查找每个函数的反向。

f(x) = 5x - 8
::f(x) = 5x-8

First write the function using “ y =” notation, then interchange x and y :
::3⁄4 ̄ ̧漯B

f(x) = 5x - 8 → y = 5x - 8 → x = 5y - 8
::f(x) = 5x-8 y = 5x-8x x = 5y-8

Then isolate y :
::然后孤立 y :

\begin{aligned} x = 5 y - 8 \end{aligned}

\begin{aligned} x + 8 = 5 y \end{aligned}

\begin{aligned} y = \frac{1}{5} x + \frac{8}{5} \end{aligned}

f(x) = x ³
:fx) = x3

Follow the same process as part 'a':
::遵循相同的程序作为“a”部分:

First write the function using “ y= ”:	f ( x ) = x ³
	y = x ³

Now interchange x and y	x = y ³
Now isolate y :	$\begin{aligned} y = \sqrt[3]{x} \end{aligned}$

Because of the definition of inverse, the graphs of inverses are reflections across the line y = x . The graph below shows $\begin{aligned} t (x) = \frac{5}{9} (x - 32) \end{aligned}$ and $\begin{aligned} f (x) = \frac{9}{5} x + 32 \end{aligned}$ on the same graph, along with the reflection line y = x .
::由于反向定义,反向图是横线y=x的反射图。下图在同一图中显示t(x)=59(x-32)和f(x)=95x+32,以及反射线y=x。

* A note about graphing with software or a graphing calculator: if you look at the graph above, you can see that the lines are reflections over the line y = x . However, if you do not view the graph in a window that shows equal scales of the x- and y- axes, the graph may not look like this.
::* 关于用软件或图形计算器绘制图的注释:如果您查看上面的图表,您可以看到这些线条是线y=x的反射。然而,如果您在显示 X 和 y 轴等比例的窗口中不查看图形,则该图可能不会像这样。

Example 4
::例4

Use composition of functions to determine if f ( x ) = 2 x + 3 and g(x) = 3 x - 2 are inverses.
::使用函数的构成来确定 f( x) = 2x + 3 和 g( x) = 3x - 2 是反向的。

The functions are not inverses.
::函数不是反向的。

We only need to check one of the compositions:
::我们只需要检查其中的其中一种成分:

$\begin{aligned} f (g (x)) = f (3 x - 2) \to 2 (3 x - 2) + 3 \to 6 x - 4 + 3 \to 6 x - 1 \neq x \end{aligned}$
::f( g( x)) = f( 3x-2) =2( 3x-2) =2( 3x-2) +3_6x-2) +3_6x-4+3_6x- 1xx

Example 5
::例5

If $\begin{aligned} f (x) = \sqrt{x + 12} \end{aligned}$ , find $\begin{aligned} f^{- 1} (x) \end{aligned}$ and determine whether f(x) is invertible.
::如果 f( x) =x+12, 则查找 f- 1 (x) 并确定 f( x) 是否不可逆。

$\begin{aligned} f (x) = \sqrt{x + 12} \to y = \sqrt{x + 12} \end{aligned}$
:xx) =x+12y=x+12

$\begin{aligned} x = \sqrt{y + 12} \end{aligned}$
::x=y+12 x=y+12

$\begin{aligned} x^{2} = (\sqrt{y + 12})^{2} \end{aligned}$ (square both sides)
::x2=(y+12)2 (两边平方)

$\begin{aligned} x^{2} = y + 12 \end{aligned}$ (simplify)
::x2=y+12(简化)

$\begin{aligned} x^{2} - 12 = y \end{aligned}$
::x2 - 12=y

$\begin{aligned} x^{2} - 12 = f (x) \end{aligned}$
::x2 - 12=f(x)

$\begin{aligned} f^{- 1} (x) = x^{2} - 12 \end{aligned}$
::f- 1 (x) =x2 - 12

In order for a function to be invertible, the inverse of the function must also be a function.
::为使一项职能可以倒置,该职能的反向也必须是一个职能。

The equation $\begin{aligned} f^{- 1} (x) = x^{2} - 12 \end{aligned}$ is a parabola with vertex (0, -12), it is indeed a function.
::等式- 1 (x) =x2 - 12 是带有顶点( 0, - 12) 的抛物线, 确实是一个函数。

Therefore f(x) is invertible.
::因此f(x)是不可倒置的。

Example 6
::例6

If $\begin{aligned} f (x) = (6, 4), (8, - 12), (- 2, 22), (10, - 10) \end{aligned}$ , find $\begin{aligned} f^{- 1} (x) \end{aligned}$ and determine whether f(x) is invertible.
::如果f(x) = (6) 4,(8) -12,(-2,22),(10,-10) , 找到 f-1(x) , 并确定 f(x) 是否不可倒置。

Just as when finding the inverse with an equation, exchange x and y.
::就像找到一个方程的反向, 交换 x 和 y 一样。

The ordered pairs (x, y) become (y, x)
::定购的对等(x, y)变成(y, x)

$\begin{aligned} ∴ f (x) = (6, 4), (8, - 12), (- 2, 22), (10, - 10) \to \end{aligned}$
::f(x) = (6) 4,(8) -12,(-2,22),(10,-10)

$\begin{aligned} \to f^{- 1} (x) = (4, 6), (- 12, 8), (22, - 2), (- 10, 10) \end{aligned}$
::f -1(x)=(4,6),(-12,8),(22,-2),(-10,10)

In order for a function to be invertible, the inverse of the function must also be a function.
::为使一项职能可以倒置,该职能的反向也必须是一个职能。

The point set: $\begin{aligned} f^{- 1} (x) = (4, 6), (- 12, 8), (22, - 2), (- 10, 10) \end{aligned}$ has no x terms with more than 1 associated y value, so it is also a function.
::点集: f-1(x) =( 4, 6), (- 12, 8), (22, -2), (- 10, 10) 没有超过 1 y 值的x 条件, 所以它也是一个函数。

Therefore f(x) is invertible.
::因此f(x)是不可倒置的。

Review
::回顾

State if the given functions are inverses of each other:
::如果给定的职能是相互对立的,请说明:

$\begin{aligned} g (x) = 4 - \frac{3}{2} x \to f (x) = \frac{1}{2} x + \frac{3}{2} \end{aligned}$
::g( x) = 4- 32xf( x) = 12x+32

%20%3D%20%5Cfrac%20%7B-12%20-%202n%7D%7B3%7D%20%5Cto%20f%20%3D%20%5Cfrac%7B-5%20%2B%206n%7D%7B5%7D"> $\begin{aligned} g (n) = \frac{- 12 - 2 n}{3} \to f (n) = \frac{- 5 + 6 n}{5} \end{aligned}$
::g +12-2n3f =5+6n5

%20%3D%20%5Cfrac%7B-16%20%2B%20n%7D%7B4%7D%20%5Cto%20g%20%3D%204n%2B%2016"> $\begin{aligned} f (n) = \frac{- 16 + n}{4} \to g (n) = 4 n + 16 \end{aligned}$
::f 16+n4g=4n+16

%20%3D%20(29n%20-%202)%5E3%20%5Cto%20g%20%3D%20%5Cfrac%7B4%20%2B%5Csqrt%5B3%5D%7B4n%7D%7D%7B2%7D"> $\begin{aligned} f (n) = (29 n - 2)^{3} \to g (n) = \frac{4 + \sqrt[3]{4 n}}{2} \end{aligned}$
::f = (29n-2) 3g = 4+4n32

$\begin{aligned} g (x) = - \frac{2}{x} - 1 \to f (x) = - \frac{2}{x + 1} \end{aligned}$
::g( x) 2x- 1f( x) 2x+1

Find the inverse of each function:
::查找每个函数的反向 :

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

$\begin{aligned} f (x) = 4 x \end{aligned}$
:xx)=4x

Find the inverse of each function. Then graph the function and its inverse.
::查找每个函数的反向。然后绘制函数及其反向图。

$\begin{aligned} f (x) = - 1 - \frac{1}{5} x \end{aligned}$
:xx)%1 - 15x

$\begin{aligned} g (x) = \frac{1}{x - 1} \end{aligned}$
::g(x)=1x-1

$\begin{aligned} f (x) = - 2 x^{3} + 1 \end{aligned}$
:xx)%2x3+1

Find the inverse of each function:
::查找每个函数的反向 :

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

Find the error in the following problem/solution:
::在以下问题/解决方案中查找错误 :

Given $\begin{aligned} f (x) = \frac{x - 10}{x} \end{aligned}$ find $\begin{aligned} f^{- 1} (x) \end{aligned}$ Step 1: $\begin{aligned} y = \frac{x - 10}{x} \end{aligned}$ Step 2: $\begin{aligned} x = \frac{y - 10}{x} \end{aligned}$ Step 3: $\begin{aligned} x (x) = (\frac{y - 10}{x}) x \end{aligned}$ Step 4: $\begin{aligned} x^{2} = y - 10 \end{aligned}$ Step 5: $\begin{aligned} x^{2} + 10 = y \end{aligned}$ Step 6: $\begin{aligned} f^{- 1} (x) = x^{2} + 10 \end{aligned}$
::给定 f( x) =x- 10x 查找 f- 1 (x) 步 1 : y=x- 10x 步 1: y= 10x 步 2: x= y- 10x 步 3 x (x) = (y- 10x) 步 4 x2= y- 10 步 5 : x2+10 = 6 f- 1 (x) = x2+10 步 : f- 1 (x) = x2+10

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

General

函数和反函数

Functions and Inverses ::函数和反函数

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Review ::回顾

Review (Answers) ::回顾(答复)