Find The Vertical Asymptotes Of The Function $\frac{(x-1)(x-3)^2(x+1)^2}{(x-2)(x+2)(x-1)(x+3)}$.A. $x=-3, X=-2, X=2$B. $x=-1, X=1, X=3$C. $x=-3, X=-2, X=1, X=2$D. $x=-1, X=3$

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Introduction

In mathematics, a vertical asymptote is a vertical line that a function approaches but never touches. It is an important concept in calculus and algebra, and it helps us understand the behavior of functions as they approach certain values. In this article, we will focus on finding the vertical asymptotes of rational functions, specifically the function (x−1)(x−3)2(x+1)2(x−2)(x+2)(x−1)(x+3)\frac{(x-1)(x-3)^2(x+1)^2}{(x-2)(x+2)(x-1)(x+3)}.

What are Vertical Asymptotes?

A vertical asymptote is a vertical line that a function approaches but never touches. It is a line that the function gets arbitrarily close to, but never actually reaches. Vertical asymptotes are typically found in rational functions, which are functions that can be written in the form f(x)g(x)\frac{f(x)}{g(x)}, where f(x)f(x) and g(x)g(x) are polynomials.

How to Find Vertical Asymptotes

To find the vertical asymptotes of a rational function, we need to find the values of xx that make the denominator of the function equal to zero. This is because a rational function is undefined when the denominator is equal to zero, and the function approaches infinity as the denominator approaches zero.

Step 1: Factor the Denominator

The first step in finding the vertical asymptotes of a rational function is to factor the denominator. This will help us identify the values of xx that make the denominator equal to zero.

Step 2: Set the Denominator Equal to Zero

Once we have factored the denominator, we can set it equal to zero and solve for xx. This will give us the values of xx that make the denominator equal to zero.

Step 3: Check for Common Factors

Before we can conclude that a value of xx is a vertical asymptote, we need to check if it is a common factor of the numerator and denominator. If it is, then the value of xx is not a vertical asymptote.

Finding the Vertical Asymptotes of the Given Function

Now that we have discussed the steps involved in finding the vertical asymptotes of a rational function, let's apply these steps to the given function (x−1)(x−3)2(x+1)2(x−2)(x+2)(x−1)(x+3)\frac{(x-1)(x-3)^2(x+1)^2}{(x-2)(x+2)(x-1)(x+3)}.

Step 1: Factor the Denominator

The denominator of the given function is (x−2)(x+2)(x−1)(x+3)(x-2)(x+2)(x-1)(x+3). We can factor this expression as follows:

(x−2)(x+2)(x−1)(x+3)=(x2−4)(x2−4)(x-2)(x+2)(x-1)(x+3) = (x^2 - 4)(x^2 - 4)

Step 2: Set the Denominator Equal to Zero

Now that we have factored the denominator, we can set it equal to zero and solve for xx. This will give us the values of xx that make the denominator equal to zero.

x2−4=0x^2 - 4 = 0

Step 3: Solve for xx

Solving the equation x2−4=0x^2 - 4 = 0, we get:

x2=4x^2 = 4

x=±2x = \pm 2

Step 4: Check for Common Factors

Before we can conclude that the values of xx are vertical asymptotes, we need to check if they are common factors of the numerator and denominator. If they are, then the values of xx are not vertical asymptotes.

The numerator of the given function is (x−1)(x−3)2(x+1)2(x-1)(x-3)^2(x+1)^2. We can see that the values of xx that make the denominator equal to zero are not common factors of the numerator and denominator.

Conclusion

In conclusion, the vertical asymptotes of the function (x−1)(x−3)2(x+1)2(x−2)(x+2)(x−1)(x+3)\frac{(x-1)(x-3)^2(x+1)^2}{(x-2)(x+2)(x-1)(x+3)} are x=−3,x=−2,x=2x = -3, x = -2, x = 2. These values of xx make the denominator equal to zero, and they are not common factors of the numerator and denominator.

Answer

The correct answer is:

A. x=−3,x=−2,x=2x=-3, x=-2, x=2

Final Thoughts

Finding vertical asymptotes is an important concept in mathematics, and it helps us understand the behavior of functions as they approach certain values. In this article, we discussed the steps involved in finding the vertical asymptotes of a rational function, and we applied these steps to the given function. We hope that this article has been helpful in understanding the concept of vertical asymptotes and how to find them.

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Introduction

In our previous article, we discussed the concept of vertical asymptotes and how to find them in rational functions. In this article, we will provide a Q&A guide to help you better understand the concept of vertical asymptotes and how to find them.

Q: What is a vertical asymptote?

A: A vertical asymptote is a vertical line that a function approaches but never touches. It is a line that the function gets arbitrarily close to, but never actually reaches.

Q: How do I find the vertical asymptotes of a rational function?

A: To find the vertical asymptotes of a rational function, you need to find the values of xx that make the denominator of the function equal to zero. This is because a rational function is undefined when the denominator is equal to zero, and the function approaches infinity as the denominator approaches zero.

Q: What are the steps involved in finding the vertical asymptotes of a rational function?

A: The steps involved in finding the vertical asymptotes of a rational function are:

  1. Factor the denominator.
  2. Set the denominator equal to zero and solve for xx.
  3. Check for common factors between the numerator and denominator.

Q: Why do I need to check for common factors?

A: You need to check for common factors because if a value of xx is a common factor of the numerator and denominator, then it is not a vertical asymptote.

Q: Can a rational function have more than one vertical asymptote?

A: Yes, a rational function can have more than one vertical asymptote. In fact, it is possible for a rational function to have an infinite number of vertical asymptotes.

Q: How do I know if a rational function has a vertical asymptote at a particular value of xx?

A: To determine if a rational function has a vertical asymptote at a particular value of xx, you need to check if the denominator of the function is equal to zero at that value of xx. If the denominator is equal to zero, then the function has a vertical asymptote at that value of xx.

Q: Can a rational function have a vertical asymptote at a value of xx that is not a zero of the denominator?

A: No, a rational function cannot have a vertical asymptote at a value of xx that is not a zero of the denominator. This is because a vertical asymptote is a line that the function approaches but never touches, and the function can only approach a line if the denominator is equal to zero at that value of xx.

Q: How do I find the vertical asymptotes of a rational function with a non-zero constant in the numerator?

A: To find the vertical asymptotes of a rational function with a non-zero constant in the numerator, you need to follow the same steps as before. However, you also need to check if the non-zero constant is a factor of the denominator.

Q: Can a rational function have a vertical asymptote at a value of xx that is a zero of the numerator?

A: No, a rational function cannot have a vertical asymptote at a value of xx that is a zero of the numerator. This is because a vertical asymptote is a line that the function approaches but never touches, and the function can only approach a line if the denominator is equal to zero at that value of xx.

Conclusion

In conclusion, finding vertical asymptotes is an important concept in mathematics, and it helps us understand the behavior of functions as they approach certain values. We hope that this Q&A guide has been helpful in understanding the concept of vertical asymptotes and how to find them.

Final Thoughts

Finding vertical asymptotes is an important concept in mathematics, and it helps us understand the behavior of functions as they approach certain values. In this article, we provided a Q&A guide to help you better understand the concept of vertical asymptotes and how to find them. We hope that this guide has been helpful in understanding the concept of vertical asymptotes and how to find them.