Course: 4.3 通过代数操纵函数的反向

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通过代数操纵使函数反向
If you were given a function, such as $\begin{aligned} f (x) = \frac{2 x}{x + 7} \end{aligned}$ , can you tell if the function has an inverse? Is there a way that you could find its inverse through algebraic manipulation?
::如果给您一个函数, 如 f( x) = 2xx+ 7 , 您能否辨别函数是否有反函数 ? 您能否通过代数操纵找到反函数 ?

Finding the Inverse of a Function
::查找函数的反向

An "inverse" is something that undoes a function, giving back the original argument. For example, a function such as $\begin{aligned} y = \frac{1}{3} x \end{aligned}$ has an inverse function of $\begin{aligned} y = 3 x \end{aligned}$ , since any value placed into the first function will be returned as what it originally was if it is input into the second function. In this case, it is easy to see that to "undo" multiplication by $\begin{aligned} \frac{1}{3} \end{aligned}$ , you should multiply by 3. However, in many cases it may not be easy to infer by examination what the inverse of a function is.
::“ 反向” 是取消函数的东西, 并给回原始参数。例如, y=13x 等函数具有y=3x 的反函数, 因为输入第二个函数后, 第一个函数中的任何值将返回原值。在这种情况下, 很容易看到“ 不做” 乘以 13 , 您应该乘以 3 。但是, 在许多情况下, 检验函数的反向值可能不容易推断出来。

To start, let's examine what is required for a function to have an inverse. It is important to remember that each function has an inverse relation and that this inverse relation is a function only if the original function is one-to-one. A function is one-to-one when its graph passes both the vertical and the horizontal line test . This means that every vertical and horizontal line will intersect the graph in exactly one place.
::首先,让我们检查函数反向需要什么。重要的是要记住, 每个函数都有反向关系, 只有原始函数为一对一时, 此反向关系才是一个函数。当图形通过垂直和水平线测试时, 一个函数是一对一。这意味着每个垂直和水平的线条都会将图形交叉到一个地方。

This is the graph of $\begin{aligned} f (x) = \frac{x}{x + 1} \end{aligned}$ . The graph suggests that $\begin{aligned} f \end{aligned}$ is one-to-one because it passes both the vertical and the horizontal line tests. To find the inverse of $\begin{aligned} f \end{aligned}$ , switch the $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ and solve for $\begin{aligned} y \end{aligned}$ .
::这是 f( x) =xx+1 的图形。图显示 f 是一对一, 因为它通过垂直和水平线测试。要找到 f 的反方向, 请切换 xandy 和溶解 Fory 。

First, switch $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ .
::首先,开关 x 和 y 。

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

::x=yyy+1 x=yyy+1

Next, multiply both sides by $\begin{aligned} (y + 1) \end{aligned}$ .
::其次,将两边乘以(y+1)

$\begin{aligned} (y + 1) x & = \frac{y}{y + 1} (y + 1) \\ x (y + 1) & = y \end{aligned}$

:y+1x=yyy+1(y+1)x(y+1)=y)

Then, apply the distributive property and put all the $\begin{aligned} y \end{aligned}$ terms on one side so you can pull out the $\begin{aligned} y \end{aligned}$ .
::然后,应用分配财产把所有Y条件放在一边这样你就可以拉出y。

$\begin{aligned} x y + x & = y \\ x y - y & = - x \\ y (x - 1) & = - x \end{aligned}$

::xy+x=yxy-yxy(x- 1) *x

Divide by $\begin{aligned} (x - 1) \end{aligned}$ to get $\begin{aligned} y \end{aligned}$ by itself.
::除以 (x-1) 以获得 y 本身。

$\begin{aligned} y = \frac{- x}{x - 1} \end{aligned}$

::yxx-1

Finally, multiply the right side by $\begin{aligned} \frac{- 1}{- 1} \end{aligned}$ .
::最后,右侧乘以-1-1。

$\begin{aligned} y = \frac{x}{1 - x} \end{aligned}$

::y=x1 - xx y=x1 - x

Therefore the inverse of $\begin{aligned} f \end{aligned}$ is $\begin{aligned} f^{- 1} (x) = \frac{x}{1 - x} \end{aligned}$ .
::因此f的反比是 f-1(x)=x1-x。

The symbol $\begin{aligned} f^{- 1} \end{aligned}$ is read “ $\begin{aligned} f \end{aligned}$ inverse” and is not the reciprocal of $\begin{aligned} f \end{aligned}$ .
::f-1的文号为“f 反向”,而不是f的对等。

Finding the Inverse of a Function
::查找函数的反向

1. Find the inverse of $\begin{aligned} f (x) = \frac{1}{x - 5} \end{aligned}$ algebraically.
::1. 查找 f(x) =1x-5 代数的反义。

To find the inverse algebraically, switch $\begin{aligned} f (x) \end{aligned}$ to $\begin{aligned} y \end{aligned}$ and then switch $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ .
::要找到反代数,请将 f(x) 切换到 y,然后将 x 和 y 切换到 y 。

$\begin{aligned} y & = \frac{1}{x - 5} \\ x & = \frac{1}{y - 5} \\ x (y - 5) & = 1 \\ x y - 5 x & = 1 \\ x y & = 5 x + 1 \\ y & = \frac{5 x + 1}{x} \end{aligned}$

::y=1x-5x=1y-5x(y-5)=1x5x=1xy=5x+1y=5x+1x

2. Find the inverse of $\begin{aligned} f (x) = 5 \sin^{- 1} (\frac{2}{x - 3}) \end{aligned}$
::2. 查找 f(x) = 5sin- 1 ( 2x- 3) 的反义值

$\begin{aligned} f (x) & = 5 \sin^{- 1} (\frac{2}{x - 3}) \\ x & = 5 \sin^{- 1} (\frac{2}{y - 3}) \\ \frac{x}{5} & = \sin^{- 1} (\frac{2}{y - 3}) \\ \sin \frac{x}{5} & = (\frac{2}{y - 3}) \\ (y - 3) \sin \frac{x}{5} & = 2 \\ (y - 3) & = \frac{2}{\sin \frac{x}{5}} \\ y & = \frac{2}{\sin \frac{x}{5}} + 3 \end{aligned}$

::f(x) = 5sin- 1( 2x-3) x= 5sin- 1( 2y-3) x5 = sin- 1( (2y-3) *x5 = (2y-3) -2y-3 (y-3) sin_x5= 2(y-3) = 2sin_x5y= 2sin_x5+3

3. Find the inverse of the trigonometric function $\begin{aligned} f (x) = 4 \tan^{- 1} (3 x + 4) \end{aligned}$
::3. 查找三角函数 f(x) = 4tan- 1 (3x+4) 的反差

$\begin{aligned} x & = 4 \tan^{- 1} (3 y + 4) \\ \frac{x}{4} & = \tan^{- 1} (3 y + 4) \\ \tan \frac{x}{4} & = 3 y + 4 \\ \tan \frac{x}{4} - 4 & = 3 y \\ \frac{\tan \frac{x}{4} - 4}{3} & = y \\ f^{- 1} (x) & = \frac{\tan \frac{x}{4} - 4}{3} \end{aligned}$

::x=4tan-1( 3y+4) x4=tan-1( 3y+4) x4=tan-1( 3y+4) x4=3y+4tan*x4-4=3ytan*x4-4=3ytan*x4- 43=yf- 1( x)=tanx4-43

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the inverse of a function.
::早些时候,有人要求你找出函数的反面。

Since the original function is:
::由于原职能是:

$\begin{aligned} f (x) = y = \frac{2 x}{x + 7} \end{aligned}$
::f(x) =y=2xx+7

You can first switch all of the "x" and "y" values:
::您可以首先切换所有“ x” 和“ Y” 值 :

$\begin{aligned} x = \frac{2 y}{y + 7} \end{aligned}$
::x=2yy+7x=2yy+7

You can then rearrange the equation and isolate "y":
::然后您可以重新排列方程,并分离“y”:

$\begin{aligned} x (y + 7) = 2 y \\ x y + 7 x = 2 y \\ x y - 2 y = - 7 x \\ y (x - 2) = - 7 x \\ y = \frac{- 7 x}{x - 2} \end{aligned}$

:y+7) = 2yxy+7x= 2yxy-2y_2y_7xy(x-2) =7xy=7xx-2

The inverse function is written as $\begin{aligned} f^{- 1} (x) = \frac{- 7 x}{x - 2} \end{aligned}$
::反函数写成 f-1( x)\\\\\\ 7xx-2 。

Example 2
::例2

Find the inverse of $\begin{aligned} f (x) = 2 x^{3} - 5 \end{aligned}$
::查找 f( x) =2x3- 5 的反方向

$\begin{aligned} f (x) & = 2 x^{3} - 5 \\ y & = 2 x^{3} - 5 \\ x & = 2 y^{3} - 5 \\ x + 5 & = 2 y^{3} \\ \frac{x + 5}{2} & = y^{3} \\ \sqrt[3]{\frac{x + 5}{2}} & = y \end{aligned}$

::f(x) = 2x3 - 5y= 2x3 - 5x= 2y3 - 5x= 2y3 - 5x+5= 2y3x+52=y3x+523=y

Example 3
::例3

Find the inverse of $\begin{aligned} y = \frac{1}{3} \tan^{- 1} (\frac{3}{4} x - 5) \end{aligned}$
::查找y=13tan-1( 34x- 5) 的反义

$\begin{aligned} y & = \frac{1}{3} \tan^{- 1} (\frac{3}{4} x - 5) \\ x & = \frac{1}{3} \tan^{- 1} (\frac{3}{4} y - 5) \\ 3 x & = \tan^{- 1} (\frac{3}{4} y - 5) \\ \tan (3 x) & = \frac{3}{4} y - 5 \\ \tan (3 x) + 5 & = \frac{3}{4} y \\ \frac{4 (\tan (3 x) + 5)}{3} & = y \end{aligned}$

::y=13tan-1(34x-5x=13tan-1(34y-55x)3x=tan-1(34y-5)(34y-5)(34y-5x)3x=34y-5tan(3x)+5=34y4(tan(3x)+5=y)3=y

Example 4
::例4

Find the inverse of $\begin{aligned} g (x) = 2 \sin (x - 1) + 4 \end{aligned}$
::查找 g( x) =2sin {( x- 1) +4 的反方向

$\begin{aligned} g (x) & = 2 \sin (x - 1) + 4 \\ y & = 2 \sin (x - 1) + 4 \\ x & = 2 \sin (y - 1) + 4 \\ x - 4 & = 2 \sin (y - 1) \\ \frac{x - 4}{2} & = \sin (y - 1) \\ \sin^{- 1} (\frac{x - 4}{2}) & = y - 1 \\ 1 + \sin^{- 1} (\frac{x - 4}{2}) & = y \end{aligned}$

::g(x)=2sin(x-1)+4y=2sin(x-1)+4x=2sin(y-1)+4x-4=2sin(y-1)+4x-4=2sin(y-1)x-42=sin(y-1)sin-1(x-42)=y-11+sin-1}(x-42)=y

Review
::回顾

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

$\begin{aligned} f (x) = 3 x + 5 \end{aligned}$
:xx)=3x+5

$\begin{aligned} g (x) = 0.2 x - 7 \end{aligned}$
::g(x) = 0.2x-7

$\begin{aligned} h (x) = 0.1 x^{2} \end{aligned}$
::h(x) = 0.1x2

$\begin{aligned} k (x) = 5 x + 6 \end{aligned}$
: kx)=5x+6)

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

$\begin{aligned} g (x) = (x)^{\frac{1}{3}} + 1 \end{aligned}$
::g(x)=(x)13+1

$\begin{aligned} h (x) = (x + 1)^{3} \end{aligned}$
::h(x)=(x+1)3

$\begin{aligned} k (x) = \frac{x^{2}}{3} \end{aligned}$
: kx) =x23

$\begin{aligned} f (x) = - 2 + 4 \sin^{- 1} (x + 7) \end{aligned}$
:xx)%2+4sin-1(x+7)

$\begin{aligned} g (x) = 1 + 3 \tan^{- 1} (2 x + 1) \end{aligned}$
::g(x) = 1+3tan- 1( 2x+1)

$\begin{aligned} h (x) = 4 \cos^{- 1} (3 x) \end{aligned}$
::h(x) = 4cos- 1 ( 3x)

$\begin{aligned} k (x) = - 1 \tan^{- 1} (6 x) \end{aligned}$
: kx) 1tan- 1( 6x)

$\begin{aligned} j (x) = 5 + 2 \sin^{- 1} (x + 5) \end{aligned}$
::j(x) = 5+2sin- 1 (x+5)

$\begin{aligned} m (x) = - 2 \tan (3 x + 1) \end{aligned}$
::m( x) 2tan( 3x+1)

$\begin{aligned} p (x) = 5 - 6 \sin (\frac{x}{2}) \end{aligned}$
::p(x)=5-6sin(x2)

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

通过代数操纵使函数反向

Finding the Inverse of a Function ::查找函数的反向

Finding the Inverse of a Function ::查找函数的反向

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Finding the Inverse of a Function
::查找函数的反向

Finding the Inverse of a Function
::查找函数的反向

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)