1至1 函数

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The statement “Pizza restaurants sell pizza” could be thought of as a function. It could be plotted on a graph, with different restaurants across the $\begin{align*}x\end{align*}$ -axis, and different foods the restaurant specializes in on the $\begin{align*}y\end{align*}$ -axis. Any time a pizza restaurant was input into the function, it would output “pizza” as the specialized food.
::“皮萨餐厅销售比萨”的说法可以被视为一种功能。 它可以在一张图上绘制,在x轴之间有不同的餐馆,餐厅在Y轴上有不同的特色食品。 每当一家比萨餐厅投入这个功能时,它就会将“比萨”作为专门食品输出。

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Is this pizza restaurant function a 1 to 1 function? How can we tell?
::这家比萨饼餐厅有1到1功能吗?

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1 to 1  Functions
::1至1 函数

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Consider the function  $\begin{align*}y=f(x)=x^3\end{align*}$ . We know  $\begin{align*}f(x)\end{align*}$ is a function, because applying the “vertical line” test to its graph shows that every value in the domain corresponds to only one value in the range. Is the inverse of  $\begin{align*}f(x)\end{align*}$ also a function?
::考虑函数 y=f(x)=x3. 我们知道 f(x) 是一个函数, 因为对图形应用“ 垂直线” 测试显示, 域内的每个值只对应区域中的一个值。 f(x) 的反方向是否也是函数?

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What do we mean by inverse of a function? The inverse of a function is obtained by taking its ordered pairs  $\begin{align*}(x,y)\end{align*}$ and switching the values of the independent and dependent variables to  $\begin{align*}(y,x)\end{align*}$ and plotting the results. This is the same as taking the equation of the function $\begin{align*}y=x^3\end{align*}$ , switching the variables  $\begin{align*}x\end{align*}$ and  $\begin{align*}y\end{align*}$ to get $\begin{align*}x=y^3\end{align*}$ , then solving for $\begin{align*}y\end{align*}$ . The result is $\begin{align*}y=\sqrt[3]{x}\end{align*}$ . The graphs of  $\begin{align*}f(x)=x^3\end{align*}$ and $\begin{align*}y=\sqrt[3]{x}\end{align*}$ are shown below.
::函数的反向值是指什么?函数的反向值是取其订购的对(x,y),将独立变量和依附变量的值切换到(y,x)并绘制结果。这与函数 y=x3 的方程式相同,将变量 x 和 y 切换到 x=y3,然后为y 解析。结果为 y=x3 。以下显示 f(x)=x3 和 y=x3 的图形。

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The vertical line test for the graph of  $\begin{align*}y=\sqrt[3]{x}\end{align*}$ shows that this inverse is also a function. The inverse function of $\begin{align*}f(x)\end{align*}$ is written as $\begin{align*}f^{-1}(x)=\sqrt[3]{x}\end{align*}$ , where the  $\begin{align*}f^{-1}(x)\end{align*}$ is the notation for indicating the inverse function of  $\begin{align*}f(x)\end{align*}$ (not to be mistaken as meaning the function to the -1 power).
::y=x3 图形的垂直线测试显示,此反函数也是一个函数。 f( x) 的反函数以 f- 1 (x) =x3 写入, 其中 f- 1 (x) 是表示 f( x) 的反函数的注解符( 不要误认为函数是 - 1 功率) 。

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The function  $\begin{align*}f(x)=x^3\end{align*}$ is an example of a 1 to 1 function. A function must be a 1 to 1 function in order to have an inverse that is a function.
::函数 f( x) =x3 是 1 到 1 函数的示例。 函数必须是 1 到 1 函数, 才能有一个反函数 。

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1 to 1 Property:  A function is 1 to 1 if and only if every element of its range corresponds to exactly one element of its domain.
::1至1财产:如果而且只有当其范围的每一要素与其域内一个要素完全对应时,函数为1至1。

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Note that there are many ways to write 1 to 1 such as one-to-one, 1:1, or 1-1. 
::请注意,写1至1有许多方法,例如一对一、一:1、1或1-1。

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The 1 to 1 property of a function can be visually tested by using the horizontal line test as described for the function $\begin{align*}y=x^2\end{align*}$ .
::函数的 1 至 1 属性可以通过对 y=x2 函数描述的水平线测试进行视觉测试。

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The function  $\begin{align*}y=x^2\end{align*}$ is not 1 to 1. The graph of this function is shown below:
::y=x2 函数不是 1 到 1. 此函数的图形显示如下 :

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Notice then that if we draw a horizontal line through $\begin{align*}y=x^2\end{align*}$ , the line touches more than one point. That indicates that the inverse will not be a function, here is why: If we invert the function $\begin{align*}y=x^2\end{align*}$ , the result is a graph that is a reflection over the line $\begin{align*}y=x\end{align*}$ , effectively rotating the original graph by 90 degrees. Since  $\begin{align*}x\end{align*}$ and  $\begin{align*}y\end{align*}$ have swapped, the new function fails the vertical line test.
::然后请注意,如果我们通过 y=x2 绘制水平线, 线会碰到一个以上的点。 这表明反向不会是一个函数, 这就是为什么 : 如果我们将函数 y=x2 反转, 结果是一个反射到 y=x 的图形, 将原始图形有效旋转90 度。 由于 x 和 y 已转换, 新函数将失败垂直线测试 。

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The function  $\begin{align*}y=x^2\end{align*}$ is therefore not a 1 to 1 function. A function that is not 1 to 1 will not have an inverse that is a function.
::y=x2 函数因此不是 1 到 1 函数。一个不是 1 到 1 函数的函数将没有函数的反函数 。

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You can determine graphically if a function is invertible (has an inverse) by applying the horizontal line test : draw a horizontal line through the graph of the function, if it touches more than one point, the function is not invertible.
::您可以通过水平线测试来图形化地确定函数是垂直的( 具有反向的) : 在函数的图形中绘制水平线, 如果它碰到一个以上的点, 函数是不可垂直的 。

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Take the graph of the function  $\begin{align*}f(x)=\frac{1}{3}x+2\end{align*}$ . Use a horizontal line test to verify that the function is invertible.
::选择函数 f( x) =13x+2 的图形。 使用水平线测试来验证函数是不可倒转的 。

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The graph below shows that this function is invertible. We can draw a horizontal line at any  $\begin{align*}y\end{align*}$ value, and the line will only cross  $\begin{align*}f(x)=\frac{1}{3}x+2\end{align*}$ once. The inverse function can be determined to be $\begin{align*}y=f^{-1}(x)=3(x-2)\end{align*}$ .
::下图显示此函数是不可翻转的。 我们可以在任何 y 值上绘制水平线, 而该线只横过 f( x) = 13x+2 一次。 反向函数可以被确定为 y= f- 1( x) = 3( x-2) 。

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In sum, a 1 to 1 function is invertible. That is, if we invert a 1 to 1 function, its inverse is also a function. Now that we have established what it means for a function to be invertible, we will focus on the domain and range of inverse functions.
::总而言之,一至一函数是不可倒置的。也就是说,如果我们将一至一函数倒置,则其反向也是一种函数。既然我们已经确定了函数不可倒置的含义,我们将把重点放在反向函数的域和范围上。

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Some functions are invertible if the domain is restricted. Take the function  $\begin{align*}f(x)=(x-2)^2\end{align*}$ .
::如果域被限制, 某些函数是不可倒置的。 使用函数 f( x) = ( x) = ( x-2) 2 。

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The graph of this function is a parabola and as such does not pass the horizontal line test. We need to limit the domain to one side of the parabola. Conventionally in cases like these we choose the positive side; therefore, the domain is limited to real numbers $\begin{align*}\ge 2\end{align*}$ .
::此函数的图形是一个抛物线, 因此无法通过水平线测试 。 我们需要将域限制在抛物线的一边 。 在类似情况下, 通常我们选择正面 ; 因此, 域限为真数 #% 2 。

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Examples
::实例

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Example 1
::例1

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Earlier, you were asked if " Pizza restaurants sell pizza” is a 1 to 1 function. It is a function, however, it is not a 1 to 1 function.
::早些时候,有人问您“皮萨餐厅卖披萨”是否是一比一的功能,但它不是一比一的功能。

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In order to be 1 to 1, it must be invertible, giving something like: “pizza sellers are pizza restaurants”, and that statement must also be a function.
::为了达到1比1,它必须是不可置疑的,给类似的东西:“披萨卖家是披萨餐厅”,而且这一声明也必须是一种功能。

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Since grocery stores sell pizza, and would therefore be among the outputs of the new function, but were not among the inputs of the original (which specified “pizza restaurants”), the functions are not invertible.
::由于杂货店出售比萨饼,因此将成为新职能的产出之一,但不属于原职能(具体指明“皮扎餐馆”)的投入,因此这些职能是不可忽略的。

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Example 2
::例2

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Is  $\begin{align*}g(x)=3x-2\end{align*}$ a 1 to 1 function?
::g(x)=3x-2 a 1 到 1 函数吗?

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The algebraic test for 1 to 1 functions is: if  $\begin{align*}f(a)=f(b)\end{align*}$ implies that $\begin{align*}a=b\end{align*}$ , then  $\begin{align*}f\end{align*}$ is 1 to 1.
::1至1函数的代数测试是:如果f(a)=f(b)意味着 a=b,那么f为1至1。

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$\begin{align*}\therefore\end{align*}$  if  $\begin{align*}g(a)=g(b)\rightarrow a=b\end{align*}$ , then  $\begin{align*}g(x)=3x-2\end{align*}$ is 1 to 1.
::如果 g(a) = g(b) a=b, 那么 g(x) =3x-2 是 1 到 1 。

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Test this:   $\begin{align*}g(a)=g(b)\end{align*}$
::测试此项: g(a)=g(b)

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$$\begin{align*}3a-2 &=3b-2\\ 3a &=3b\\ a &=b\end{align*}$$
::3a-2=3b-23a=3ba=b

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$\begin{align*}\therefore 3x-2\end{align*}$ is $\begin{align*}1-1\end{align*}$
::3x-2是1-1

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Example 3
::例3

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Use the horizontal line test to see if  $\begin{align*}f(x)=x^3\end{align*}$ is 1 to 1.
::使用水平线测试查看 f( x) =x3 是否为 1 到 1 。

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Graph the equation:
::方程式图解 :

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This is the parent function of the cubic function family. Each  $\begin{align*}x\end{align*}$ value has one unique $\begin{align*}y\end{align*}$ -value that is not used by any other $\begin{align*}x\end{align*}$ -element. Since that is the definition of a 1 to 1 function, this function is 1 to 1.
::这是立方函数家族的父函数。 每个 x 值有一个独特的 Y 值, 而其他任何 x 元素都没有使用。 由于这是 1 到 1 函数的定义, 此函数为 1 到 1 。

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Example 4
::例4

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Is  $\begin{align*}g(x)=|x-2|\end{align*}$ 1 to 1?
::g(x)x-21至1?

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Graph the equation:
::方程式图解 :

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This absolute value function has $\begin{align*}y\end{align*}$ -values that are paired with more than one $\begin{align*}x\end{align*}$ -value, such as (4, 2) and (0, 2). This function is not 1 to 1. Note that this function also fails the horizontal line test.
::此绝对值函数的 Y 值配有多个 X 值, 如 (4, 2) 和 (0, 2) 。 此函数不是 1 至 1. 。 请注意, 此函数也使水平线测试失败 。

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Example 5
::例5

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State the domain and range of the function and its inverse:
::状态 函数的域和范围及其反向 :

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Function: (1, 2), (2, 5), (3, 7)
::职能1,2),(2,5),(3,7)

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The inverse of this function is the set of points (2, 1), (5, 2), (7, 3)
::此函数的反向值是一组点(2、1)、5、2、7、3)

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The domain of the function is {1, 2, 3}. This is also the range of the inverse.
::函数的域为 {1, 2, 3}。 这也是反向的范围 。

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The range of the function is {2, 5, 7}. This is also the domain of the inverse.
::函数的范围为 {2, 5, 7}。 这也是反向的域 。

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The linear functions we examined previously, as well as $\begin{align*}f(x)=x^3\end{align*}$ , all had domain and range both equal to the set of all real numbers. Therefore the inverses also had domain and range equal to the set of all real numbers. Because the domain and range were the same for these functions, switching them maintained that relationship.
::我们以前检查过的线性函数以及 f(x) =x3 , 都拥有域和范围, 与所有实际数字的一组相等。 因此反向也拥有域和范围, 与所有实际数字的一组相等。 由于域和范围与这些函数相同, 转换后它们保持了这种关系 。

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Also, as we found above, the function  $\begin{align*}y=x^2\end{align*}$ is not 1 to 1, and hence it is not invertible. That is, if we invert it, the resulting relation is not a function. We can change this situation if we define the domain of the function in a more limited way. Let $\begin{align*}f(x)\end{align*}$ be a function defined as follows: $\begin{align*}f(x)=x^2\end{align*}$ , with domain limited to real numbers ≥ 0. Then the inverse of the function is the square root function: $\begin{align*}f^{-1}(x)=\sqrt{x}\end{align*}$
::此外,正如我们上文所发现的那样,y=x2 函数不是一对一,因此它不是不可倒置的。也就是说,如果我们颠倒它,由此产生的关系不是函数。如果我们以更有限的方式定义函数的域,我们可以改变这种情况。让我们f(x) 是一个定义如下的函数: f(x) =x2, 域限为真数 = 0。然后函数的反面是平方根函数 : f-1(x) =x

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Review
::回顾

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1. Describe the 1 to 1 Horizontal Line Test
   ::描述 1 到 1 水平线测试
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2. Describe the 1 to 1 Algebraic Test
   ::描述 1 到 1 代数测试

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For #3-5, state whether or not the function is 1 to 1.
::对于 # 3-5, 请说明该函数是否为 1 到 1 。

3. $\begin{align*}(3,28),(4,29),(4,30),(6,31)\end{align*}$
4. $\begin{align*}(4,5),(9,6),(7,8),(23,5)\end{align*}$
5. $\begin{align*}(8,18),(33,4),(5,16),(7,19)\end{align*}$

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For #6-7, for the relation  be a 1 to 1 function,  $\begin{align*}X\end{align*}$ cannot be what values?
::对于#6-7, 如果关系是 1 到 1 函数, X 不能是什么值 ?

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6. $\begin{align*}(9,12),(35,6),(7,18),(12,X)\end{align*}$
   :9,12),(35,6),(7,18),(12,X)
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7. $\begin{align*}(20,21),(21,14),(110,112),(X,7)\end{align*}$
   :20,21,(21,14),(110,112),(X,7))

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For #8-12, state whether or not the function is 1 to 1.
::对于# 8-12, 请说明该函数是否为 1 到 1 。

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8. $\begin{align*}f(x)=x^2\end{align*}$
   :xx)=x2
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9. $\begin{align*}f(x)=x^3\end{align*}$
   :xx)=x3
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10. $\begin{align*}f(x)=\frac{1}{x}\end{align*}$
    :xx)=1x
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11. $\begin{align*}f(x)=x^n-x, n>0\end{align*}$
    :xx)=xn-x,n>0
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12. $\begin{align*}x=y^2+2\end{align*}$
    ::x=y2+2 x=y2+2

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For #13-15, determine if the relations are  functions, 1 to 1 functions or neither.
::对于第13-15号,确定关系是否为职能、1至1职能或两者兼而有之。

13. 
14. 
15. 

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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 。

 

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Resources
::资源