Section: 函数的逆函数 | 1.12 函数的反义

Section outline

Functions are commonly known as rules that take inputs and produce outputs. An i nverse function does exactly the reverse, undoing what the original function does. How can you tell if two functions are inverses?
::函数通常被称为接收输入并产生输出的规则。反向函数正好相反, 推翻原始函数所做的。您如何知道两个函数是否反向 ?

Finding Inverses of Functions
::查找函数的反侧

A function is written as $\begin{aligned} f (x) \end{aligned}$ and its inverse is written as $\begin{aligned} f^{- 1} (x) \end{aligned}$ . A common misconception is to see the -1 and interpret it as an exponent and write $\begin{aligned} \frac{1}{f (x)} \end{aligned}$ , but this is not correct. Instead, $\begin{aligned} f^{- 1} (x) \end{aligned}$ should be viewed as a new function from the range of $\begin{aligned} f (x) \end{aligned}$ back to the domain.
::函数是 f(x) 写入的, 其反写是 f- 1(x) 。常见的误解是将 -1 视为引号, 并将其解释为推号, 并写入 1f(x) , 但这不正确。相反, f- 1(x) 应被视为从 f(x) 回溯到域范围的新函数。

It is important to see the cycle that starts with $\begin{aligned} x \end{aligned}$ , becomes $\begin{aligned} y \end{aligned}$ and then goes back to $\begin{aligned} x \end{aligned}$ . In order for two functions to truly be inverses of each other, this cycle must hold algebraically.
::重要的是要看到以 x 开始的周期, 变成 y 然后返回到 x 。为了让两个函数真正反向, 这个周期必须保持代数。

$\begin{aligned} f (f^{- 1} (x)) = x \end{aligned}$ and $\begin{aligned} f^{- 1} (f (x)) = x \end{aligned}$
::f( f- 1 (x)) =x 和 f- 1 (f(x)) =x

When given a function there are two steps to take to find its inverse. In the original function, first switch the variables $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ . Next, solve the function for $\begin{aligned} y \end{aligned}$ . This will give you the inverse function. After finding the inverse, it is important to check both directions of compositions to make sure that together the function and its inverse produce the value $\begin{aligned} x \end{aligned}$ . In other words, verify that $\begin{aligned} f (f^{- 1} (x)) = x \end{aligned}$ and $\begin{aligned} f^{- 1} (f (x)) = x \end{aligned}$ .
::当给定一个函数时, 有两个步骤可以找到它的反向值。在原始函数中, 先切换变量 x 和 y 。下一步, 解决 y 的函数。这将给您提供反向函数。在查找反向函数后, 需要检查组合的双向, 以确保函数及其反向生成值 x。换句话说, 验证 f( f- 1 (x) =x 和 f- 1 (f(x)) =x 。

Graphically, inverses are reflections across the line $\begin{aligned} y = x \end{aligned}$ . Below you see inverses $\begin{aligned} y = e^{x} \end{aligned}$ and $\begin{aligned} y = \ln x \end{aligned}$ . Notice how the $\begin{aligned} (x, y) \end{aligned}$ coordinates in one graph become $\begin{aligned} (y, x) \end{aligned}$ coordinates in the other graph.
::图形化时,反向是横跨 y=x 线的反射。下面您可以看到 y=ex 和 y= ln= x 。注意一个图形中的( x,y) 坐标是如何在另一个图形中变成 (y, x) 坐标的。

In order to decide whether an inverse function is also actually a function you can use the vertical line test on the inverse function like usual. You can also use the horizontal line test on the original function. The horizontal line test is exactly like the vertical line test except the lines simply travel horizontally.
::为了决定一个反向函数是否也是一个函数, 您可以像往常一样在反向函数上使用垂直线测试。您也可以在原始函数上使用水平线测试。水平线测试与垂直线测试完全相同, 除了线条只是水平运行。

Examples
::实例

Example 1
::例1

Earlier, you were asked how you can tell that two functions are inverses. You can tell that two functions are inverses if each undoes the other, always leaving the original $\begin{aligned} x \end{aligned}$ .
::早些时候,有人问您如何辨别两个函数是反向的。您可以辨别两个函数是反向的,如果每个函数推翻另一个函数,则两个函数是反向的,总是离开原来的 x 。

Example 2
::例2

Find the inverse, then verify the inverse algebraically. $\begin{aligned} f (x) = y = (x + 1)^{2} + 4 \end{aligned}$
::查找反向, 然后验证反代数。 f( x)=y=( x+1) 2+4

To find the inverse, switch $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ then solve for $\begin{aligned} y \end{aligned}$ .
::寻找反向, 切换 x 和 y , 然后为 y 解答。

\begin{aligned} x & = (y + 1)^{2} + 4 \\ x - 4 & = (y + 1)^{2} \\ \pm \sqrt{x - 4} & = y + 1 \\ - 1 \pm \sqrt{x - 4} & = y = f^{- 1} (x) \end{aligned}

::x= (y+1) 2+4x- 4= (y+1) 2x- 4= (y+1) 2x- 4=y+1- 1x- 4=y=f- 1(x)

To verify algebraically, you must show $\begin{aligned} x = f (f^{- 1} (x)) = f^{- 1} (f (x)) \end{aligned}$ :
::要校验代数, 您必须显示 x=f( f- 1 (x)) = f- 1 (f(x) :

\begin{aligned} f (f^{- 1} (x)) & = f (- 1 \pm \sqrt{x - 4}) \\ = ((- 1 \pm \sqrt{x - 4}) + 1)^{2} + 4 \\ = {(\pm \sqrt{x - 4})}^{2} + 4 \\ = x - 4 + 4 = x \end{aligned}

::f( f- 1 (xx)) = f( 1-x- 4) = (- 1x- 4) = (- 1x- 4) +1) 2+4 = (x- 4) 2+4= (x- 4) 2+4= x-4+4=x

\begin{aligned} f^{- 1} (f (x)) & = f^{- 1} ((x + 1)^{2} + 4) \\ = - 1 \pm \sqrt{((x + 1)^{2} + 4) - 4} \\ = - 1 \pm \sqrt{(x + 1)^{2}} \\ = - 1 + x + 1 = x \end{aligned}

::f- 1 (f(x)) = f- 1 ((x+1) 2+4)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

As you can see from the graph, the $\begin{aligned} \pm \end{aligned}$ causes the inverse to be a relation instead of a function. This can be observed in the graph because the original function does not pass the horizontal line test and the inverse does not pass the vertical line test.
::从图中可以看到, ° 使反向成为关系而不是函数。这可以在图中观察到, 因为原函数没有通过水平线测试, 而反向则没有通过垂直线测试。

Example 3
::例3

Find the inverse of the function and then verify that $\begin{aligned} x = f (f^{- 1} (x)) = f^{- 1} (f (x)) \end{aligned}$ .
::查找函数的反向,然后核实 x=f(f-1(x)) = f- 1(f(x) 。

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

Sometimes it is quite challenging to switch $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ and then solve for $\begin{aligned} y \end{aligned}$ . You must be careful with your algebra.
::有时转换 x 和 y 然后为 y 解决问题相当困难。您必须小心代数。

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

::x=y+1y-1x(y- 1)=y+1xy-x=y+1xy-y=x1y(x-1)=x1y=x1y=x1x-x-1)

This function turns out to be its own inverse. Since they are identical, you only need to show that $\begin{aligned} x = f (f^{- 1} (x)) \end{aligned}$ .
::3⁄4 ̄ ̧漯B

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

Example 4
::例4

What is the inverse of $\begin{aligned} f (x) = y = \sin x \end{aligned}$ ?
::F(x)=y=sinx的反义值是什么?

The sine function does not pass the horizontal line test and so its true inverse is not a function.
::正弦函数不会通过水平线测试, 因此它的真实反向不是一个函数。

However, if you restrict the domain to just the part of the $\begin{aligned} x \end{aligned}$ -axis between $\begin{aligned} - \frac{π}{2} \end{aligned}$ and $\begin{aligned} \frac{π}{2} \end{aligned}$ then it will pass the horizontal line test and the inverse will be a function.
::但是,如果您将域限制在% 2 和% 2 之间的 X 轴部分,则它会通过水平线测试,而反向则是一个函数。

The inverse of the sine function is called the arcsine function, $\begin{aligned} f (x) = \sin^{- 1} (x) \end{aligned}$ , and is shown in black. It is truncated so that it only inverts a part of the whole sine wave. You will study periodic functions and their inverses in more detail later.
::正弦函数的反义称为arcsine 函数, f(x) =sin- 1(x) , 以黑色显示。它被缩短, 因而它只能反转整个正弦波的一部分。您以后将更详细地研究周期函数及其反函数。

Example 5
::例5

Determine if $\begin{aligned} f (x) = \frac{3}{7} x - 21 \end{aligned}$ and $\begin{aligned} g (x) = \frac{7}{3} x + 21 \end{aligned}$ are inverses of one another.
::确定 f( x) = 37x- 21 和 g( x) = 73x+21 是否是彼此的反义。

Even though $\begin{aligned} f (x) = \frac{3}{7} x - 21 \end{aligned}$ and $\begin{aligned} g (x) = \frac{7}{3} x + 21 \end{aligned}$ have some inverted pieces, they are not inverses of each other. In order to show this, you must show that the composition does not simplify to $\begin{aligned} x \end{aligned}$ .
::尽管 f( x) = 37x- 21 和 g( x) = 73x+21 有一些反向的片段, 但两者并非反向的。要显示这一点, 您必须显示组成不简化为 x 。

$\begin{aligned} \frac{3}{7} (\frac{7}{3} x + 21) - 21 = x + 9 - 21 = x - 12 \neq x \end{aligned}$
::37(73x+21)-21=x+9-21=x-12x

Summary

Inverse functions undo the actions of the original functions, and are denoted as $\begin{aligned} f^{- 1} (x) . \end{aligned}$
::反向函数撤消原始函数的动作,并被指为 f-1(x) 。

To find the inverse of a function, switch the variables x and y, and then solve the function for y.
::要查找函数的反向,请切换变量 x 和 y,然后为 y 解析函数。

To verify if two functions $\begin{aligned} f (x) \end{aligned}$ and $\begin{aligned} g (x) \end{aligned}$ are inverses, check that $\begin{aligned} f (g (x)) = x \end{aligned}$ and $\begin{aligned} g (f (x)) = x . \end{aligned}$
::要验证两个函数 f( x) 和 g( x) 是否为反函数, 请检查 f( g( x)) =x 和 g( f( x)) =x 。

Graphically, inverses are reflections across the line $\begin{aligned} y = x, \end{aligned}$ with x and y coordinates switching places.
::图形化,反向是横跨 y=x 线的反射,有 x 和 y 坐标切换位置。

Review
::回顾

Consider $\begin{aligned} f (x) = x^{3} \end{aligned}$ .
::考虑 f(x) =x3 。

1. Sketch $\begin{aligned} f (x) \end{aligned}$ and $\begin{aligned} f^{- 1} (x) \end{aligned}$ .
::1. Spetch f(x) 和 f- 1(x) 。

2. Find $\begin{aligned} f^{- 1} (x) \end{aligned}$ algebraically. It is actually a function?
::2. 查找 f-1(x) 代数。它实际上是一个函数吗 ?

3. Verify algebraically that $\begin{aligned} f (x) \end{aligned}$ and $\begin{aligned} f^{- 1} (x) \end{aligned}$ are inverses.
::3. 从代数上核实f(x)和f-1(x)是反向的。

Consider $\begin{aligned} g (x) = \sqrt{x} \end{aligned}$ .
::考虑 g(x)=x。

4. Sketch $\begin{aligned} g (x) \end{aligned}$ and $\begin{aligned} g^{- 1} (x) \end{aligned}$ .
::4. Spetch g(x) 和 g- 1(x) 。

5. Find $\begin{aligned} g^{- 1} (x) \end{aligned}$ algebraically. It is actually a function?
::5. 查找 g-1(x) 代数。它实际上是一个函数吗 ?

6. Verify algebraically that $\begin{aligned} g (x) \end{aligned}$ and $\begin{aligned} g^{- 1} (x) \end{aligned}$ are inverses.
::6. 核实g(x)和g-1(x)是反向的代数。

Consider $\begin{aligned} h (x) = | x | \end{aligned}$ .
::考虑 h(x) 。

7. Sketch $\begin{aligned} h (x) \end{aligned}$ and $\begin{aligned} h^{- 1} (x) \end{aligned}$ .
::7. Spetch h(x) 和 h-1(x) 。

8. Find $\begin{aligned} h^{- 1} (x) \end{aligned}$ algebraically. It is actually a function?
::8. 查找 h-1(x) 代数。它实际上是一个函数吗 ?

9. Verify graphically that $\begin{aligned} h (x) \end{aligned}$ and $\begin{aligned} h^{- 1} (x) \end{aligned}$ are inverses.
::9. 以图形方式验证 h(x) 和 h- 1(x) 是反向的。

Consider $\begin{aligned} j (x) = 2 x - 5 \end{aligned}$ .
::考虑j(x)=2x-5。

10. Sketch $\begin{aligned} j (x) \end{aligned}$ and $\begin{aligned} j^{- 1} (x) \end{aligned}$ .
::10. Spletch j(x) 和 j-1(x) 。

11. Find $\begin{aligned} j^{- 1} (x) \end{aligned}$ algebraically. It is actually a function?
::11. 查找 j-1(x) 代数。它实际上是一个函数吗 ?

12. Verify algebraically that $\begin{aligned} j (x) \end{aligned}$ and $\begin{aligned} j^{- 1} (x) \end{aligned}$ are inverses.
::12. 从代数上核实j(x)和j-1(x)是反向的。

13. Use the horizontal line test to determine whether or not the inverse of $\begin{aligned} f (x) = x^{3} - 2 x^{2} + 1 \end{aligned}$ is also a function.
::13. 使用水平线测试来确定 f( x) =x3 - 2x2+1 的反向是否也是一种函数。

14. Are $\begin{aligned} g (x) = \ln (x + 1) \end{aligned}$ and $\begin{aligned} h (x) = e^{x - 1} \end{aligned}$ inverses? Explain.
::14. g(x) = ln(x+1) 和 h(x) = ex- 1 反义吗? 解释。

15. If you were given a table of values for a function, how could you create a table of values for the inverse of the function?
::15. 如果给您一个函数的数值表,您如何为函数的反向创建数值表?

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

Finding Inverses of Functions ::查找函数的反侧

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)