Course: 2.7 合理函数中的空洞

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理性函数中的空洞

In a function like

\begin{aligned} f (x) = \frac{(3 x + 1) (x - 1)}{(x - 1)} \end{aligned}

, you should note that the factor

\begin{aligned} (x - 1) \end{aligned}

clearly cancels leaving only

\begin{aligned} 3 x - 1 \end{aligned}

. This appears to be a regular line. What happens to this line at

\begin{aligned} x = 1 \end{aligned}

Holes and Rational Functions
::孔和合理函数

A hole on a graph looks like a hollow circle. It represents the fact that the function approaches the point, but is not actually defined on that precise $\begin{aligned} x \end{aligned}$ value.
::图形上的空洞看起来像空圆。它表示函数接近点, 但实际上没有根据精确的 x 值定义。

Take a look at the graph of the following equation :
::查看以下方程式的图示:

$\begin{aligned} f (x) = (2 x + 2) \cdot \frac{(x + \frac{1}{2})}{(x + \frac{1}{2})} \end{aligned}$
::f(xx) = (2x+2) = (x+12) (x+12)

The reason why this function is not defined at $\begin{aligned} - \frac{1}{2} \end{aligned}$ is because $\begin{aligned} - \frac{1}{2} \end{aligned}$ is not in the domain of the function. As you can see, $\begin{aligned} f (- \frac{1}{2}) \end{aligned}$ is undefined because it makes the denominator of the rational part of the function zero which makes the whole function undefined. Also notice that once the factors are canceled/removed then you are left with a normal function which in this case is $\begin{aligned} 2 x + 2 \end{aligned}$ . The hole in this situation is at $\begin{aligned} (- \frac{1}{2}, 1) \end{aligned}$ because after removing the factors that cancel, $\begin{aligned} f (- \frac{1}{2}) = 1 \end{aligned}$ .
::1- 12 没有定义此函数的原因是- 12 不在该函数的域内。正如你可以看到的那样, f(- 12) 尚未定义, 因为它使函数零的理性部分的分母使得整个函数没有定义。另外请注意, 元素取消/ 移动后, 您将留下一个正常函数, 在本案中为 2x+2. 。此情况下的空洞是 (- 12, 1) , 因为在删除取消系数后, f(- 12)= 1 。

This is the essence of dealing with holes in rational functions . You should cancel what you can and graph the function like normal making sure to note what $\begin{aligned} x \end{aligned}$ values make the function undefined. Once the function is graphed without holes go back and insert the hollow circles indicating what $\begin{aligned} x \end{aligned}$ values are removed from the domain. This is why holes are called removable discontinuities.
::这是处理理性函数中空洞的精髓。您应该取消您能够的功能, 并像普通函数一样将函数图形化, 以确保注意到什么 x 值使函数未定义。一旦该函数被图形化而没有空洞, 就会返回, 插入空心圆, 以显示从域中移除的 x 值。这就是为什么洞被称为可移动的不连续性。

Watch the first part of this video and focus on holes in rational equations.
::观看这段视频的第一部分, 并关注理性方程式的漏洞。

Examples
::实例

Example 1
::例1

Earlier, you were asked what happens to the equation $\begin{aligned} f (x) = \frac{(3 x + 1) (x - 1)}{(x - 1)} \end{aligned}$ at $\begin{aligned} x = 1 \end{aligned}$ . Since this function that is not defined at $\begin{aligned} x = 1 \end{aligned}$ there is a removable discontinuity that is represented as a hollow circle on the graph. Otherwise the function behaves precisely as $\begin{aligned} 3 x + 1 \end{aligned}$ .
::早些时候,有人问您方程式 f( x) =( 3x+1)( x- 1)( x- 1)( x- 1)( x- 1)) 在 x= 1 上会发生什么情况。由于此函数在 x=1 时没有定义, 图形上有一个可移动的不连续状态, 以空圆表示。否则, 此函数的运行将精确为 3x+1 。

Example 2
::例2

Graph the following rational function and identify any removable discontinuities.
::绘制以下合理函数图,并标明任何可移动的不连续状态。

$\begin{aligned} f (x) = \frac{- x^{3} + 3 x^{2} + 2 x - 4}{x - 1} \end{aligned}$
:xx)x3+3x2+2x-4x-1

This function requires some algebra to change it so that the removable factors become obvious. You should suspect that $\begin{aligned} (x - 1) \end{aligned}$ is a factor of the numerator and try polynomial or synthetic division to factor. When you do, the function becomes:
::此函数需要某些代数才能更改它, 使可移动因素变得显而易见。您应该怀疑 (x- 1) 是分子的一个系数, 并尝试多分子或合成分数作为系数。当您修改代数时, 该函数会变成 :

$\begin{aligned} f (x) = \frac{(- x^{2} + 2 x + 4) (x - 1)}{(x - 1)} \end{aligned}$
:xx) = (-x2+2x+4)(x-1)(x-1) = (-x2+2x+4)(x-1)(x-1)

The removable discontinuity occurs at (1, 5).
::可移动的不连续性发生在(1, 5)时。

Example 3
::例3

Graph the following rational function and identify any removable discontinuities.
::绘制以下合理函数图,并标明任何可移动的不连续状态。

$\begin{aligned} f (x) = \frac{x^{6} - 6 x^{5} + 5 x^{4} + 27 x^{3} - 48 x^{2} - 9 x + 54}{x^{3} - 7 x - 6} \end{aligned}$
:x) =x6- 6x5+5x4+27x3- 48x2- 9x+54x3- 7x-6)

This is probably one of the most challenging rational expressions with only holes that people ever try to graph by hand. There are multiple ways to start, but a good habit to get into is to factor everything you possibly can initially. The denominator seems less complicated with possible factors $\begin{aligned} (x \pm 1), (x \pm 2), (x \pm 3), (x \pm 6) \end{aligned}$ . Using polynomial division , you will find the denominator becomes:
::这可能是最具挑战性的理性表达方式之一, 只有人们曾经尝试用手写出洞洞。有多种方法可以启动。但一个好的习惯是首先考虑你所能考虑的所有因素。分母与可能的因素( x+1, (xx%2), (x+3), (x+6)相比似乎不那么复杂。使用多元分母, 你会发现分母会变成 :

$\begin{aligned} f (x) = \frac{x^{6} - 6 x^{5} + 5 x^{4} + 27 x^{3} - 48 x^{2} - 9 x + 54}{(x + 1) (x + 2) (x - 3)} \end{aligned}$
::f(x) =x6- 6x5+5x4+27x3- 482- 9x54(x+1)(x+2)(x-3)

The factors of the denominator are strong hints as to the factors of the numerator so use polynomial division and try each. When you fully factor the numerator you will have:
::分母的因子是分子因子的强烈提示,所以要使用多等分数来尝试每一个。当您充分乘以该分子时,您将拥有:

$\begin{aligned} f (x) = \frac{(x^{3} - 6 x^{2} + 12 x - 9) (x + 1) (x + 2) (x - 3)}{(x + 1) (x + 2) (x - 3)} \end{aligned}$
:xx) = (x3- 6x2+12x- 9) (x+1)(x+2)(x-2)(x-3)(x+1)(x+2)(x-3)

Note the factors that cancel $\begin{aligned} (x = - 1, - 2, 3) \end{aligned}$ and then work with the cubic function that remains.
::注意取消(x1,-2,3)和与剩余立方函数一起工作的因素。

$\begin{aligned} f (x) = x^{3} - 6 x^{2} + 12 x - 9 \end{aligned}$
:x) =x3- 6x2+12x- 9

At this point it is probably reasonable to make a table and plot points to get a sense of where this cubic function lives. You also could notice that the coefficients are almost of the pattern 1 3 3 1 which is the binomial expansion. By separating the -9 into -8 -1 you can factor the first four terms .
::此时此刻,也许可以合理选择一个表格和绘图点, 以了解此立方函数的寿命。您也可以注意到, 系数几乎是二进制扩展模式 13 3 3 1 。通过将 - 9 和 - 8 - 1 分隔开来, 您可以将前四个条件考虑在内。

$\begin{aligned} f (x) = x^{3} - 6 x^{2} + 12 x - 8 - 1 = (x - 2)^{3} - 1 \end{aligned}$
:x) =x3 - 6x2+12x - 8-8 - 1=(x-2) 3 - 1

This is a cubic function that has been shifted right by two units and down one unit.
::这是一个立方函数, 由两个单位向右移动, 一个单位向下移动。

Note that there are two holes that do not fit in the graph window. When this happens you still need to note where they would appear given a properly sized window. To do this, substitute the invalid $\begin{aligned} x \end{aligned}$ values: $\begin{aligned} x = - 1, - 2, 3 \end{aligned}$ into the factored cubic that remained after canceling.
::请注意, 图形窗口中有两个不匹配的洞。发生这种情况时, 您仍需要注意它们会在哪里出现, 给定一个适当大小的窗口。要做到这一点, 请将无效的 x 值: x1, ~2, 3 替换为取消后留下的计数立方体。

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

Holes: (3, 0); (-1, -28); (-2, -65)
::洞3,0);(1,28);(2,65)

Example 4
::例4

Without graphing, identify the location of the holes of the following function.
::在不绘制图形的情况下,标明下列函数的孔的位置。

$\begin{aligned} f (x) = \frac{x^{3} + 4 x^{2} + x - 6}{x^{2} + 5 x + 6} \end{aligned}$
::f(x) =x3+4x2+x-6x2+5x+6

First factor everything. Then, identify the $\begin{aligned} x \end{aligned}$ values that make the denominator zero and use those values to find the exact location of the holes.
::首先是所有因素。然后, 确定使分母为零的 x 值, 并使用这些值找到洞的确切位置。

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

Holes: (-3, -4); (-2, -3)
::洞3-3-4);(2-3)

Example 5
::例5

What is a possible equation for the following rational function?
::以下合理功能的可能方程式是什么?

The function seems to be a line with a removable discontinuity at (1, -1). The line is has 1 and $\begin{aligned} y \end{aligned}$ - intercept of -2 and so has the equation:
::函数似乎与 1, - 1 的可移动不连续性一致。该直线为 - 2 的 1 和 Y 界面, 等式也如此 :

$\begin{aligned} f (x) = x - 2 \end{aligned}$
:xx)=x-2

The removable discontinuity must not allow the $\begin{aligned} x \end{aligned}$ to be 1 which implies that it is of the form $\begin{aligned} \frac{x - 1}{x - 1} \end{aligned}$ . Therefore , the function is:
::可移动的不连续性不得允许 x 1 表示其为表x-1x-1。因此,该函数是:

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

Summary

A hole on a graph is a hollow circle, representing that the function approaches the point but is not defined at that precise value.
::图形上的空洞是空圆,表示函数接近点,但没有以该精确值定义。

Holes occur in rational functions when the function is undefined at a specific value.
::当函数未按特定值定义时,在合理函数中会出现空洞。

To graph a function with holes, cancel out the removable factors and graph the remaining function, noting the values that make the function undefined.
::要绘制带有空洞的函数图,请取消可移动系数,并绘制剩余函数图,同时注意使函数未定义的值。

After graphing the function without holes, insert hollow circles at the undefined values to indicate the holes.
::在绘制无空洞的函数图后,插入空心圆圈,以未定义的值表示空洞。

Review
::回顾

1. How do you find the holes of a rational function?
::1. 如何找到理性功能的漏洞?

2. What’s the difference between a hole and a removable discontinuity?
::2. 洞与可移动的不连续之间有什么区别?

3. If you see a hollow circle on a graph, what does that mean?
::3. 如果在图表上看到空圆,那是什么意思?

Without graphing, identify the location of the holes of the following functions.
::在不绘制图表的情况下,标明下列函数的孔的位置。

4. $\begin{aligned} f (x) = \frac{x^{2} + 3 x - 4}{x - 1} \end{aligned}$
::4. f(x)=x2+3x-4x-1

5. $\begin{aligned} g (x) = \frac{x^{2} + 8 x + 15}{x + 3} \end{aligned}$
::5. g(x)=x2+8x+15x+3

6. $\begin{aligned} h (x) = \frac{x^{3} - 2 x^{2} - 5 x + 6}{x^{2} + x - 2} \end{aligned}$
::6. h(x)=x3-2x2-5x+6x2+x-2

7. $\begin{aligned} k (x) = \frac{x^{2} + x - 2}{x^{2} - 3 x + 2} \end{aligned}$
::7. k(x) =x2+x-2x2-2x2-3x+2

8. $\begin{aligned} j (x) = \frac{x^{3} + 4 x^{2} - 17 x - 60}{x^{2} - 9} \end{aligned}$
::8. j(x)=x3+4x2-17x-60x2-9

9. $\begin{aligned} f (x) = \frac{x^{3} + 4 x^{2} - 17 x - 60}{x^{2} - 5 x + 4} \end{aligned}$
::9. f(x)=x3+4x2-17x-60x2-5x+4

10. $\begin{aligned} g (x) = \frac{x^{3} - 4 x^{2} - 19 x - 14}{x^{2} - 8 x + 7} \end{aligned}$
::10. g(x)=x3-4x2-19x-14x2-8x+7

11. What is a possible equation for the following rational function?
::11. 以下合理功能的可能等式是什么?

12. What is a possible equation for the following rational function?
::12. 以下合理功能的可能等式是什么?

Sketch the following rational functions.
::伸展以下的理性功能。

13. $\begin{aligned} f (x) = \frac{x^{3} + 4 x^{2} - 17 x - 60}{x^{2} - x - 12} \end{aligned}$
::13. f(x)=x3+4x2-17x-60x2-x-12

14. $\begin{aligned} g (x) = \frac{x^{3} + 4 x^{2} - 17 x - 60}{x^{2} + 8 x + 15} \end{aligned}$
::14. g(x)=x3+4x2-17x-60x2+8x+15

15. $\begin{aligned} h (x) = \frac{x^{3} - 4 x^{2} - 19 x - 14}{x^{2} - 6 x - 7} \end{aligned}$
::15. h(x) =x3-4x2-19x-14x2-6-7

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

理性函数中的空洞

Holes and Rational Functions ::孔和合理函数

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)