反反函数

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Most of the functions from previous concepts have been polynomial functions, or rational functions, or functions generated using a finite number of terms, involving only the operations of addition, subtraction, multiplication, division, and raising to integer or fractional powers. These functions are algebraic functions, and when there is an inverse, the inverse is also an algebraic function. Functions that are not algebraic functions are called transcendental functions, i.e., functions that “transcend” algebra in the sense that they cannot be expressed in terms of a finite set of the algebraic operations of addition/subtraction, multiplication/division, and power/root. The three groups of transcendental functions that will be discussed in subsequent concepts, along with their inverses, are the “elementary” transcendentals:
::先前概念中的大多数功能是多元函数,或合理函数,或使用一定数量术语产生的功能,仅涉及增量、减量、倍增、分化和向整数或分数功率的操作。这些函数是代数函数,如果反之,则反之亦为代数函数。非代数函数被称为超度函数,即“转换”代数函数,即不能用一组有限的增量/增量、倍增量/倍增量和功率/根等代数函数表示的“转换”代数函数。

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• Exponential/inverse exponential (logarithmic) functions,
  ::指数/反向指数(对数)函数,
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• Trigonometric/inverse trigonometric functions, and
  ::三角/逆三角函数,以及
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• Hyperbolic/inverse hyperbolic functions
  ::双曲/反双曲函数

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Have you heard of hyperbolic functions, or any other transcendental functions? Because the functions in each of the above groups has an inverse useful for solving certain types of problems, you should know why a function inverse is important, and how to use it. See if you can state, before you read further, the basic property of an inverse, and the condition that allows a function to have an inverse.
::您听说过双曲函数或其他超常函数吗? 因为上述各组的函数对于解决某些类型的问题有反向的用处, 您应该知道为什么反向函数是重要的, 以及如何使用它。 看看您能否在阅读反向函数之前说明反向函数的基本属性, 以及允许反向函数的条件 。

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Inverses
::逆数

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In this section we will look at the following topics relating to :
::在本节中,我们将探讨以下与下列方面有关的议题:

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• What does it mean to have an inverse?
  ::反悔是什么意思?
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• When does a function have an inverse?
  ::函数何时反转 ?
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• Finding the inverse of a function
  ::查找函数的反向
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• Graphs of Inverse Functions
  ::反反函数图
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• of Inverse Functions
  ::反向函数

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What does it mean to have an inverse?
::反悔是什么意思?

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If a function $\begin{align*}f(x)\end{align*}$  can be reversed in such a way that the input of the function becomes the output and the output becomes an input, and the resulting reversed relationship is itself a function $\begin{align*}h(x)\end{align*}$ , then  $\begin{align*}h(x)\end{align*}$ is the inverse of $\begin{align*}f(x)\end{align*}$ .  This relationship can be written as:
::如果函数 f(x) 能够倒转, 使函数的输入变成输出, 输出变成输入, 所产生的反向关系本身就是函数 h(x), 那么h(x) 是 f(x) 的反义。 此关系可以写为 :

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$\begin{align*}f \circ h = f(h(x)) = h \circ f = x\end{align*}$
::fh=f(h(x))=hf=x

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The inverse function  $\begin{align*}h\end{align*}$ is often written as $\begin{align*}f^{-1}\end{align*}$ , which is not to be confused as $\begin{align*}\frac{1}{f}\end{align*}$ . In general, $\begin{align*}f^{-1} \neq \frac{1}{f}\end{align*}$ .
::反函数 h 通常以 f-1 写成,不得混淆为 1f 。一般而言, f-1 1f 。

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If a function has an inverse, it is said to be invertible.
::如果一个函数反之,据说是倒置的。

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Take the following two functions $\begin{align*}f(x) = 2x+3\end{align*}$  and $\begin{align*}h(x) = \frac{x-3}{2}\end{align*}$  . They are inverses of each other since
::以下两个函数 f( x) = 2x+3 和 h( x) =x- 32 。 它们自此反相

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$\begin{align*}f \circ h = f(h(x)) = 2 \left [ \frac{x-3}{2} \right ] + 3 = x - 3 + 3 =x\end{align*}$ , and $\begin{align*}h \circ f = h(f(x)) = \frac{(2x+3)-3}{2} = \frac{2x}{x} =x\end{align*}$
::fh=f(h(xx))=2[x- 32]+3=x3=3+3=x, hf=h(f(x))=(2x+3)-32=2xx=x

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Thus
::因此,

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$\begin{align*}f \circ h = h \circ f = x\end{align*}$ ,
::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}

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and $\begin{align*}f\end{align*}$  and $\begin{align*}h\end{align*}$  are inverses of each other. The function $\begin{align*}f(x) = 2x+3\end{align*}$  is invertible.
::f 和 h 是彼此的反义。函数 f( x) = 2x+3 是不可逆的。

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When does a function have an inverse?
::函数何时反转 ?

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A function $\begin{align*}f(x)\end{align*}$  has an inverse if it is one-to-one in its domain, or if its derivative is either $\begin{align*}f^\prime (x) > 0\end{align*}$  or $\begin{align*}f^\prime (x) < 0\end{align*}$ .
::如果函数f(x)在其域内为一对一,或如果其衍生物为f`(x)>0或f`(x)<0,则函数f(x)具有反向。

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A function must be one-to-one (1-to-1) to have an inverse, i.e., to be invertible. A function is said to be a one-to-one function if each output is associated with only one single input. For example, $\begin{align*}f(x) = x^2\end{align*}$  assigns the output 9 for both 3 and -3 and therefore is not a one-to-one function.
::函数必须是一对一(1至-1),才能产生反向的函数,即不可倒置。如果每项产出仅与一个输入相关,则该函数称为一对一。例如,f(x)=x2为3和-3指定了输出9,因此不是一对一的函数。

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Another way to define one-to-one is: the function $\begin{align*}f(x)\end{align*}$  is one-to-one in a domain $\begin{align*}D\end{align*}$  if $\begin{align*}f(a) \neq f(b)\end{align*}$  whenever $\begin{align*}a \neq b\end{align*}$ .
::定义一对一的另一种方法是:函数f(x)是域D的一对一,如果f(a)f(b)在ab时。

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There is an easy method to check if a function is one-to-one called the horizontal line test: if a horizontal line drawn across the graph intersects the graph at only one point on the graph, then the function is one-to-one; otherwise, it is not. Notice in the figure below that the graph of $\begin{align*}y=x^2\end{align*}$  is not one-to-one since the horizontal line intersects the graph more than once. But the function $\begin{align*}y=x^3\end{align*}$  is a one-to-one function because the graph meets the horizontal line only once.
::有一种简单的方法可以检查函数是否为一对一,称为水平线测试:如果在图形中划过一条水平线时将图形相交,则该函数为一对一;否则则不是。下图中注意,y=x2的图形不是一对一,因为水平线将图形多次相交。但是,y=x3的函数是一个一对一的函数,因为图形只与水平线相交一次。

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Note that if a function $\begin{align*}f(x)\end{align*}$  is always increasing or always decreasing over its domain, then a horizontal line will cut through this graph at one point only. Then $\begin{align*}f(x)\end{align*}$  in this case is a one-to-one function and thus has an inverse. So if we can find a way to prove that a function is constantly increasing or decreasing, then it is invertible .  The way to do this is to look at the derivative of the function. From previous chapters, you have learned that if $\begin{align*}f^\prime (x) > 0\end{align*}$  then $\begin{align*}f\end{align*}$  must be increasing; and if $\begin{align*}f^\prime (x) < 0\end{align*}$  then  $\begin{align*}f\end{align*}$ must be decreasing. This means that for some functions the determination of the one-to-one property can be expressed by using the derivative of the function.
::请注意, 如果函数 f( x) 在其域上总是在增加或总是在减少, 那么水平线就会在某个点上切穿此图。 然后, f( x) 将是一个一对一的函数, 因而有一个反向。 因此, 如果我们能找到一种方法来证明函数在不断增加或减少, 那么它是不可逆的。 这样做的方法是查看函数的衍生物。 从前几章中, 您已经了解到, 如果 f` (x) > 0, 那么f 必须增加; 如果 f ' (x) < 0, 那么f 必须减少。 这意味着, 对于某些函数来说, 一对一属性的确定可以通过使用函数的衍生物来表示 。

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Using the information above, determine whether these functions have inverses, i.e., are invertible:
::使用上述信息,确定这些函数是否有反效果,即:

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1. $\begin{align*}f(x) = |x|\end{align*}$
   :fx)
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2. $\begin{align*}h(x) = x^{\frac{1}{2}}\end{align*}$
   ::h(x)=x12
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3. $\begin{align*}f(x) = 3x^5 + 2x +1\end{align*}$
   ::f(x)=3x5+2x+1

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For  $\begin{align*}f(x) = |x|\end{align*}$  and  $\begin{align*}h(x) = x^{\frac{1}{2}}\end{align*}$ , it is best to graph both functions and draw on each a horizontal line. As you can see from the graphs,  $\begin{align*}f(x) = |x|\end{align*}$ is not one-to-one since the horizontal line intersects it at two points. The function $\begin{align*}h(x) = x^{\frac{1}{2}}\end{align*}$  however, is indeed one-to-one since only one point is intersected by the horizontal line.
::对于 f( x) x 和 h( x) =x12 , 最好在每一水平线上同时绘制函数和绘制一个水平线。 从图形中可以看到, f( x) x 不是一对一, 因为水平线在两个点交叉。 但是, 函数 h( x) =x12 确实是一对一, 因为水平线只交叉一个点。

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For,  $\begin{align*}f(x) = 3x^5 + 2x +1\end{align*}$ , taking the derivative, we find that $\begin{align*}f^{\prime} (x) = 15x^4 + 2 > 0\end{align*}$  for all $\begin{align*}x\end{align*}$ . We therefore conclude that $\begin{align*}f(x)\end{align*}$  is one-to-one and invertible. Keep in mind that it may not be easy to find the inverse (try it!), but we still know that it is indeed invertible.
::对于 f( x) = 3x5+2x+1, 使用衍生物, 我们发现所有 x 的 f`( x) = 15x4+2>0 。 因此, 我们得出结论, f( x) 是一对一, 是不可倒置的 。 记住要找到反义可能不容易( 尝试它! ) , 但我们仍然知道它确实是不可倒置的 。

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Finding the inverse of a one-to-one function
::查找一对一函数的反向

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To find the inverse of a one-to-one function, perform the following steps if possible:
::为查找一对一函数的反差,如有可能,执行下列步骤:

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1. Solve for the independent variable $\begin{align*}x\end{align*}$  in terms of the dependent variable $\begin{align*}y\end{align*}$
   ::以依附变量 y 为独立的变量 x 解决
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2. Interchange $\begin{align*}x\end{align*}$  and $\begin{align*}y\end{align*}$ . 
   ::互换x和y。

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The resulting formula is the inverse $\begin{align*}y = f^{-1} (x)\end{align*}$ .
::结果的公式是反y=f-1(x) 。

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Of course, it is not always easy or possible to perform the first step.
::当然,采取第一步并不总是容易或可能的。

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Note that it is also acceptable to perform Step 2 above first, then to solve for the dependent variable $\begin{align*}y\end{align*}$ .
::请注意,也可以接受先执行以上步骤2,然后解决依附变量y。

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Graphs of Inverse Functions
::反反函数图

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What is the relationship between the graphs of $\begin{align*}f\end{align*}$  and $\begin{align*}f^{-1}\end{align*}$ ? If the point $\begin{align*}(a, b)\end{align*}$  is on the graph of $\begin{align*}f(x)\end{align*}$ , then from the definition of the inverse, the point $\begin{align*}(b, a)\end{align*}$  is on the graph of $\begin{align*}f^{-1}(x)\end{align*}$ . In other words, when we reverse the coordinates of a point on the graph of $\begin{align*}f(x)\end{align*}$  we automatically get a point on the graph of $\begin{align*}f^{-1}(x)\end{align*}$ . Examination of the resulting graphs will show that the function and its inverse are reflections of one another about the line $\begin{align*}y=x\end{align*}$ . That is, each is a mirror image of the other about the line $\begin{align*}y = x\end{align*}$ . The figure below shows an example of $\begin{align*}y = x^2\end{align*}$  and, when the domain is restricted to $\begin{align*}x \ge 0\end{align*}$ , its inverse $\begin{align*}y = \sqrt{x}\end{align*}$  and how they are reflected about $\begin{align*}y=x\end{align*}$ .
::f 和 f - 1 的图形之间的关系是什么? 如果 点 (a, b) 在 f (x) 的图形上, 然后从 y (x) 的定义中, 点 (b, a) 在 f - 1 (x) 的图形上。 换句话说, 当我们翻转 f (x) 图形上的一个点的坐标时, 我们自动在 f - 1 (x) 的图形上找到一个点。 对由此生成的图形的检查将显示, 该函数及其反向是 y =x 的反射。 也就是说, 每个是 y =x 的反射镜。 下图显示一个 y= x 的示例, 当域限制为 x 10 时, 其 y=x 的反向 和 y=x 的反射方式 。

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It is important to note that for the function $\begin{align*}f(x) = x^2\end{align*}$  to have an inverse, we must restrict its domain to $\begin{align*}0 \le x < \infty\end{align*}$ , since that is the domain in which the function is increasing.
::必须指出,要使f(x)=x2的函数反转,我们必须将其域限为 0x,因为这是函数增加的领域。

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Continuity and Differentiability of Inverse Functions
::反向函数的连续性和差异性

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Since the graph of a one-to-one function and its inverse are reflections of one another about the line $\begin{align*}y = x\end{align*}$  it would be safe to say that if the function $\begin{align*}f\end{align*}$  has no breaks (no discontinuities) then $\begin{align*}f^{-1}\end{align*}$  will not have breaks either. This implies that if $\begin{align*}f\end{align*}$  is continuous on the domain $\begin{align*}D\end{align*}$ , then its inverse $\begin{align*}f^{-1}\end{align*}$  is continuous on the range $\begin{align*}R\end{align*}$  of $\begin{align*}f\end{align*}$ . For example, if $\begin{align*}f(x) = \sqrt{x}\end{align*}$ , then its domain is $\begin{align*}x \ge 0\end{align*}$  and its range is $\begin{align*}y \ge 0\end{align*}$ . This means that $\begin{align*}f(x)\end{align*}$  is continuous for all $\begin{align*}x \ge 0\end{align*}$ . The inverse of $\begin{align*}f(x)\end{align*}$  is $\begin{align*}f^{-1}(x) = x^2\end{align*}$ , where its domain is all $\begin{align*}x > 0\end{align*}$  and its range is $\begin{align*}y \ge 0\end{align*}$ . We conclude that if $\begin{align*}f\end{align*}$  is a function with domain $\begin{align*}D\end{align*}$  and range $\begin{align*}R\end{align*}$  and it is continuous and one-to-one on $\begin{align*}D\end{align*}$ , then its inverse $\begin{align*}f^{-1}\end{align*}$  is continuous and one-to-one on the range $\begin{align*}R\end{align*}$  of  $\begin{align*}f\end{align*}$ .
::由于一对一函数的图形及其反向是横线 y=x 的反射,所以可以安全地说,如果函数 f 没有折断(不不不连续), f-1 也不会断断。这意味着,如果 f 连续在 D 域上,那么它的反向 f-1 在 f 范围上是连续的。例如,f(x) =x,它的域是 x=0,它的域是 y=0。这意味着 f(x) 对所有 x=0 是连续的。 f(x) 的反方向是 f- 1 (x) =x 2, 其域是 全部 x > 0 , 其范围是 y* 0 。我们的结论是, 如果 f 是 域 D 的函数, R 范围是连续的, D 范围是一对一, 那么它的反方向 f-1 是连续的, f 范围是 R 的一对一 。

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Continuity of the Inverse Function
::反向函数的连续性

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If $\begin{align*}f(x)\end{align*}$  is a function with domain $\begin{align*}D\end{align*}$  and range $\begin{align*}R\end{align*}$ , and it is continuous and one-to-one on $\begin{align*}D\end{align*}$ , then its inverse $\begin{align*}f^{-1}(x)\end{align*}$  is continuous and one-to-one on the range $\begin{align*}R\end{align*}$  of $\begin{align*}f\end{align*}$ .
::如果 f(x) 是域 D 和 范围 R 的函数, 并且是连续的, 在 D 上一对一, 那么它的反面 f-1(x) 是连续的, 在 R 的 R 上一对一 。

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The differentiability of the inverse follows from the differentiability of the function $\begin{align*}f(x)\end{align*}$ .
::反向的可区别性源于函数f(x)的可区别性。

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Differentiability of the Inverse Function
::Inverse 函数的可区分性

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Suppose $\begin{align*}f(x)\end{align*}$  is a function with a domain  $\begin{align*}D\end{align*}$ that is an open interval, and a range $\begin{align*}R\end{align*}$ , and an inverse $\begin{align*}f^{-1}(x)\end{align*}$ .
::假设f(x) 是一个函数, 域D是开放间隔, 范围R 和反向f- 1(x) 。

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If $\begin{align*}f(x)\end{align*}$  is differentiable at $\begin{align*}f^{-1}(x)\end{align*}$  and $\begin{align*}f^{\prime}(f^{-1}(x)) \neq 0\end{align*}$ , then $\begin{align*}f^{-1}(x)\end{align*}$  is differentiable at $\begin{align*}x\end{align*}$  and the following differentiation formula holds:
::如果f(x)在 f- 1(x) 和 f}(f- 1(x)) 上是可变的,则f- 1(x) 在 x 上是可变的,并持有下列区别公式:

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$$\begin{align*}\frac{d}{dx} [f^{-1}(x)] = \frac{1}{f^{\prime}(f^{-1}(x))} \ or \ \frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}\end{align*}$$
::ddx[f- 1(x)]=1f_(f- 1(x))或uddx=1dxdy

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The same differentiation formula holds, if  $\begin{align*}f(x)\end{align*}$  is either $\begin{align*}f^{\prime}(x) > 0\end{align*}$  or $\begin{align*}f^{\prime}(x) < 0\end{align*}$  on its domain.
::如果f(x)在其域内是 f`(x) >0 或 f`(x) <0 ,则相同的区别公式保持不变。

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Earlier, the polynomial function $\begin{align*}f(x) = 3x^5 + 2x +1\end{align*}$  was shown to be invertible.
::早些时候,多位函数 f( x) = 3x5+2x+1 被显示为不可翻转。

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Now let's show that the function is differentiable and find the derivative of its inverse.
::现在让我们来证明这个函数是不同的 并且找到它的反向衍生物

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Since  $\begin{align*}f^{\prime} (x) = 15x^4 + 2 >0\end{align*}$ for all $\begin{align*}x \in R, f^{-1} (x)\end{align*}$  is differentiable at all values of $\begin{align*}x\end{align*}$ . To find the derivative of  $\begin{align*}f^{-1}\end{align*}$ , if we let  "> $\begin{align*}x = f(y)\end{align*}$ , then
::由于 f{( x) = 15x4+2>0 对所有 x}\\\\ r, f- 1( x) 在所有 x 值上都是不同的。 如果我们让 x= f, 找到 f- 1 的衍生物, 那么

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%20%3D%203y%5E5%20%2B%202y%20%2B%201"> $\begin{align*}x = f(y) = 3y^5 + 2y + 1\end{align*}$ .
::x=f=3y5+2y+1;

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So
::苏

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$\begin{align*}\frac{dx}{dy} = 15y^4 + 2\end{align*}$
::dxdy=15y4+2 (dxdy=15y4+2)

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and
::和

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$\begin{align*}\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}} = \frac{1}{15y^4 + 2}\end{align*}$ .
::dydx=1dxdy=115y4+2。

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Since we are unable to solve for $\begin{align*}y\end{align*}$  in terms of $\begin{align*}x\end{align*}$ , we leave the answer above in terms of  $\begin{align*}y\end{align*}$ . Another way of solving the problem is to use :
::由于我们无法从x的角度解决y的问题,所以我们从y的角度留下上面的答案。

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Since
::自

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$\begin{align*}x = 3y^5 + 2y +1\end{align*}$ ,
::x=3y5+2y+1, x=3y5+2y+1,

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differentiating implicitly gives,
::隐含的区别规定,

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$$\begin{align*}\frac{d}{dx} [x] &= \frac{d}{dx} [3y^5 +2y+1]\\ 1 & = (15y^4 +2)\frac{dy}{dx}\end{align*}$$
::ddx[x]=ddx[3y5+2y+1]1=(15y4+2)uddx

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Solving for  $\begin{align*}\frac{dy}{dx}\end{align*}$ we finally obtain
::解决我们最终获得的

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$\begin{align*}\frac{dy}{dx} = \frac{1}{15y^4 + 2}\end{align*}$ ,
::dydx=115.y4+2, dydx=115.y4+2

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which is the same result.
::这是相同的结果。

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Examples
::实例

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Example 1
::例1

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Earlier, you were asked to name different types of  functions have inverses. 
::早些时候,有人要求你列出不同类型具有反作用的函数。

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Other than the exponential, logarithmic, and trigonometric functions, you may not have heard of any other transcendental functions. But, there are quite a few that are used extensively in science and engineering applications, including the hyperbolic functions which have properties similar to the trigonometric functions but are made up of exponential functions. Other transcendental functions have names such as Bessel functions and Hankel functions, and arise in specialized areas. The typical math student may never encounter these.
::除了指数函数、对数函数和三角函数之外,你可能没有听说过任何其他超常函数。但是,在科学和工程应用中,有相当多的超曲函数被广泛使用,包括超曲函数,这些函数的属性与三角函数相似,但由指数函数组成。其他超函数有贝塞尔函数和汉克尔函数等名称,并出现在专门领域。典型数学学生可能从未遇到过这些功能。

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The basic property of the function inverse when it exists is: $\begin{align*}f^{-1} \circ f = x = f \circ f^{-1}\end{align*}$ .
::函数存在时的反基本属性为 : f-1\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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For a function, $\begin{align*}f(x)\end{align*}$ , to have an inverse , $\begin{align*}f^{-1}(x)\end{align*}$ , the function must be one-to-one.
::函数 f(x) 的反函数为 f-1(x),函数必须是一对一。

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Example 2
::例2

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Find the inverse of $\begin{align*}f(x) = \sqrt{4x+1}\end{align*}$ .
::查找 f( x) =4x+1 的反义。

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First of all, we know the function is one-to-one over the domain $\begin{align*}x \ge \frac{1}{4}\end{align*}$ , and therefore it has an inverse.
::首先,我们知道这个函数在 x14 的域上是一对一, 因此它反之亦然 。

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From the discussion above, we first solve for $\begin{align*}x\end{align*}$ :
::从上面的讨论,我们首先解决x问题:

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$$\begin{align*}y & = \sqrt{4x+1}\\ y^2 & = 4x+1\\ x & = \frac{y^2-1}{4}\end{align*}$$
::y=4x+1y2=4x+1x=y2-14

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Next, perform the variable interchange  $\begin{align*}x\end{align*}$ and  $\begin{align*}y\end{align*}$ to determine the inverse
::下一步,执行变量交换 x 和 y 以决定反向

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$\begin{align*}y = \frac{x^2 - 1}{4}\end{align*}$ .
::y=x2 -14。 y=x2 -14。

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Replacing  $\begin{align*}y = f^{-1} (x)\end{align*}$ ,
::替换y=f- 1(x),

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$\begin{align*}f^{-1} (x) = \frac{x^2-1}{4}\end{align*}$
::f- 1 (x) =x2 - 14

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which is the inverse of the original function $\begin{align*}f(x) = \sqrt{4x+1}\end{align*}$ .
::等于原函数f(x)=4x+1的反义。

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Example 3
::例3

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Given  the function $\begin{align*}f(x) = 3x^3 + 9x + 4\end{align*}$ . Show that the function invertible. Find the value of $\begin{align*}\frac{d}{dx} f^{-1} \Big |_{y=4}\end{align*}$
::根据 函数 f( x) = 3x3+9x+4, 显示函数不可翻转。 查找 ddxf- 1\\\\\\\\ y=4 的值

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Since $\begin{align*}f^{\prime} (x) = 9x^2+ 9 > 0\end{align*}$ , the function has an inverse.  To find the inverse function, let "> $\begin{align*}x=f(y)\end{align*}$ , so that $\begin{align*}x=3y^3+9y+4\end{align*}$ .
::自 fä( x) = 9x2+9>0 以来, 该函数有一个反函数。 要找到反函数, let x=f, 所以 x=3y3+9y+4 。

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Using implicit differentiation yields:
::使用隐含差异收益率:

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$\begin{align*}dx = 9y^2 dy + 9 dy \Rightarrow 1 = 9(y^2+1) \frac{dy}{dx}\end{align*}$ .
::dx=9y2dy+9dy1=9(y2+1)dydx。

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This means $\begin{align*}\frac{d}{dx} f^{-1} = \frac{1}{9(y^2+1)}\end{align*}$ , and
::这意味着 ddxf- 1=19(y2+1) 和

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$\begin{align*}\frac{d}{dx} f^{-1} \Big |_{y=4} = \frac{1}{9(16+1)} = \frac{1}{153}\end{align*}$ .
::ddxf- 1_y=4=19(16+1)=1153。

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Review
::回顾

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For  #1 - 3, find the inverse function of $\begin{align*}f\end{align*}$  and verify that $\begin{align*}f \circ f^{-1} = f^{-1} \circ f = x\end{align*}$ .
::对于 # 1 - 3, 找到 f 的反函数, 并校验 ff - 1 = f - 1 f=x 。

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1. $\begin{align*}f(x) = 3x+1\end{align*}$
   :xx)=3x+1
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2. $\begin{align*}\sqrt[3]{x}\end{align*}$
   ::x3x3
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3. $\begin{align*}f(x) = \frac{x-1}{3}\end{align*}$
   :xx)=x-13)

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For  #4 - 6, use the horizontal line test to verify whether the following functions have inverse.
::对于 # 4-6, 使用水平线测试来验证以下函数是否反向 。

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4. $\begin{align*}h(x) = \frac{4-x}{6}\end{align*}$
   ::h(x)=4 - x6
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5. $\begin{align*}g(x) = |x+4| - |x-4|\end{align*}$
   ::g( x) x+4x- 4
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6. $\begin{align*}f(x) = -2x \sqrt{16-x^2}\end{align*}$
   :xx)%2x16-x2

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For  #7 - 8, use the functions $\begin{align*}f(x) = x+4\end{align*}$  and $\begin{align*}g(x) = 2x-5\end{align*}$  to find the specified functions.
::对于 # 7 - 8, 使用 f( x) =x+4 和 g( x) = 2x - 5 函数查找指定的函数 。

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7. $\begin{align*}g^{-1} \circ f^{-1}\end{align*}$
   ::-1-1 -1个
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8. $\begin{align*}(f \circ g)^{-1}\end{align*}$
   :fg)-1

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For #9 - 10, show that $\begin{align*}f\end{align*}$  is monotonic (invertible) on the given interval (and therefore has an inverse.)
::对于 # 9 - 10, 显示 F 是给定间隔的单音( 不可忽略) (因此反音 ) 。

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9. $\begin{align*}f(x) = (x-5)^2, [5, \infty)\end{align*}$
   :xx)=(x-5)2,[5,]
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10. $\begin{align*}f(x) = \cos x, \left [ 0, \frac{\pi}{2} \right ]\end{align*}$
    ::f(x)=cosx,[0],[%2]

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For #11-15 find the inverse, if it exists, and determine the domain and range of the inverse.
::对于#11-15, 发现反向, 如果它存在, 并确定反向的域和范围 。

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11. $\begin{align*}f(x) = \frac{6x-1}{3x+7}\end{align*}$
    :xx)=6x-13x+7
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12. $\begin{align*}f(x) = x^2 - 4x + 8\end{align*}$  for $\begin{align*}x \ge 2\end{align*}$
    ::x% 2 的 f( x) =x2 - 4x+8
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13. $\begin{align*}f(x) = \frac{13x}{11x+5}\end{align*}$
    :xx)=13x11x+5
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14. $\begin{align*}f(x) = x^7 - 2\end{align*}$
    :xx)=x7-2)
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15. $\begin{align*}f(x) = \frac{1+\sqrt{x}}{1-\sqrt{x}}\end{align*}$
    :xx)=1+x1-xx

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Review (Answers)
::回顾(答复)

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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 。