At Which Value(s) Of X X X Does The Graph Of The Function F ( X F(x F ( X ] Have A Vertical Asymptote? Check All That Apply. F ( X ) = X ( X − 4 ) ( X + 2 ) F(x)=\frac{x}{(x-4)(x+2)} F ( X ) = ( X − 4 ) ( X + 2 ) X ​ A. X = 4 X=4 X = 4 B. X = − 2 X=-2 X = − 2 C. X = − 4 X=-4 X = − 4 D. X = 2 X=2 X = 2 E.

Introduction

In mathematics, a vertical asymptote is a vertical line that a function approaches but never touches. It occurs when the denominator of a rational function is equal to zero, causing the function to become undefined at that point. In this article, we will explore the concept of vertical asymptotes and determine the values of $x$ for which the graph of the function $F(x) = \frac{x}{(x-4)(x+2)}$ has a vertical asymptote.

What are Vertical Asymptotes?

A vertical asymptote is a vertical line that a function approaches but never touches. It occurs when the denominator of a rational function is equal to zero, causing the function to become undefined at that point. In other words, a vertical asymptote is a line that the function gets arbitrarily close to but never crosses.

How to Find Vertical Asymptotes

To find the vertical asymptotes of a rational function, we need to find the values of $x$ that make the denominator equal to zero. We can do this by setting the denominator equal to zero and solving for $x$. In the case of the function $F(x) = \frac{x}{(x-4)(x+2)}$, we need to find the values of $x$ that make the expression $(x-4)(x+2)$ equal to zero.

Solving for $x$

To solve for $x$, we can set the expression $(x-4)(x+2)$ equal to zero and solve for $x$. We can do this by using the distributive property to expand the expression:

$$(x-4)(x+2) = x^2 - 4x + 2x - 8$$

Simplifying the expression, we get:

$$x^2 - 2x - 8 = 0$$

We can solve this quadratic equation by factoring or using the quadratic formula. Factoring, we get:

$$(x - 4)(x + 2) = 0$$

This tells us that either $x - 4 = 0$ or $x + 2 = 0$. Solving for $x$, we get:

$$x = 4 \text{ or } x = -2$$

Conclusion

In conclusion, the graph of the function $F(x) = \frac{x}{(x-4)(x+2)}$ has vertical asymptotes at $x = 4$ and $x = -2$. These values of $x$ make the denominator of the function equal to zero, causing the function to become undefined at those points.

Answer

Based on our analysis, the correct answers are:

• A. $x = 4$
• B. $x = -2$

The other options are incorrect because they do not make the denominator of the function equal to zero.

Discussion

This problem requires the student to understand the concept of vertical asymptotes and how to find them. It also requires the student to be able to solve quadratic equations and factor expressions. The student should be able to explain why the function has vertical asymptotes at $x = 4$ and $x = -2$ and why the other options are incorrect.

Additional Examples

Here are a few additional examples of rational functions with vertical asymptotes:

• $F(x) = \frac{x}{x^2 - 4}$
• $F(x) = \frac{x}{x^2 + 4}$
• $F(x) = \frac{x}{x^2 - 9}$

In each of these examples, the student should be able to find the values of $x$ that make the denominator equal to zero and determine the vertical asymptotes of the function.

Conclusion

Q: What is a vertical asymptote?

A: A vertical asymptote is a vertical line that a function approaches but never touches. It occurs when the denominator of a rational function is equal to zero, causing the function to become undefined at that point.

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 $x$ that make the denominator equal to zero. You can do this by setting the denominator equal to zero and solving for $x$.

Q: What is the difference between a vertical asymptote and a hole in a graph?

A: A vertical asymptote is a vertical line that a function approaches but never touches, while a hole in a graph is a point where the function is undefined but the graph passes through that point. Holes occur when there is a common factor in the numerator and denominator of a rational function.

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

A: Yes, a rational function can have more than one vertical asymptote. This occurs when the denominator of the function is a product of two or more factors, each of which can be equal to zero.

Q: How do I determine if a rational function has a vertical asymptote at a particular value of $x$?

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

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

A: No, a rational function cannot have a vertical asymptote at a value of $x$ that is not a zero of the denominator. A vertical asymptote occurs when the denominator is equal to zero, so if the denominator is not equal to zero at a particular value of $x$, then the function does not have a vertical asymptote at that value of $x$.

Q: How do I graph a rational function with vertical asymptotes?

A: To graph a rational function with vertical asymptotes, you need to identify the vertical asymptotes and plot them on the graph. You can do this by finding the values of $x$ that make the denominator equal to zero and plotting the corresponding vertical lines on the graph.

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

A: No, a rational function cannot have a vertical asymptote at a value of $x$ that is a zero of the numerator. A vertical asymptote occurs when the denominator is equal to zero, so if the numerator is equal to zero at a particular value of $x$, then the function does not have a vertical asymptote at that value of $x$.

Q: How do I determine if a rational function has a hole in its graph?

A: To determine if a rational function has a hole in its graph, you need to check if there is a common factor in the numerator and denominator of the function. If there is a common factor, then the function has a hole in its graph at the value of $x$ that makes the common factor equal to zero.

Q: Can a rational function have both vertical asymptotes and holes in its graph?

A: Yes, a rational function can have both vertical asymptotes and holes in its graph. This occurs when the denominator of the function is a product of two or more factors, each of which can be equal to zero, and there is a common factor in the numerator and denominator of the function.

Conclusion

In conclusion, vertical asymptotes are an important concept in mathematics that can be used to analyze the behavior of rational functions. By understanding how to find vertical asymptotes, students can gain a deeper understanding of the properties of rational functions and how to work with them.