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 + 2 ) ( X − 1 ) F(x)=\frac{x}{(x+2)(x-1)} F ( X ) = ( X + 2 ) ( X − 1 ) X ​ A. X = − 2 X=-2 X = − 2 B. X = 1 X=1 X = 1 C. X = − 1 X=-1 X = − 1 D. X = 10 X=10 X = 10 E.

Introduction

In mathematics, a vertical asymptote is a vertical line that a function approaches but never touches. It is a point where the function's graph becomes infinite or undefined. Rational functions, which are the ratio of two polynomials, can have vertical asymptotes at certain values of the variable. In this article, we will explore the concept of vertical asymptotes and determine the values of $x$ for which the graph of the function $F(x) = \frac{x}{(x+2)(x-1)}$ has a vertical asymptote.

What are Vertical Asymptotes?

A vertical asymptote is a vertical line that a function approaches but never touches. It is a point where the function's graph becomes infinite or undefined. In other words, a vertical asymptote is a line that the function gets arbitrarily close to but never crosses. Vertical asymptotes are typically denoted by the symbol $\infty$ and are often represented as a vertical line on a graph.

How to Find Vertical Asymptotes

To find the vertical asymptotes of a rational function, we need to look for the values of the variable 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. Therefore, we can find the vertical asymptotes by setting the denominator equal to zero and solving for the variable.

The Function $F(x) = \frac{x}{(x+2)(x-1)}$

The given function is $F(x) = \frac{x}{(x+2)(x-1)}$. To find the vertical asymptotes of this function, we need to set the denominator equal to zero and solve for $x$. The denominator of the function is $(x+2)(x-1)$, which is equal to zero when $x+2=0$ or $x-1=0$.

Solving for $x$

Let's solve for $x$ by setting each factor equal to zero:

• $x+2=0 \Rightarrow x=-2$
• $x-1=0 \Rightarrow x=1$

Therefore, the values of $x$ for which the denominator of the function is equal to zero are $x=-2$ and $x=1$.

Conclusion

In conclusion, the graph of the function $F(x) = \frac{x}{(x+2)(x-1)}$ has a vertical asymptote at $x=-2$ and $x=1$. These values of $x$ make the denominator of the function equal to zero, which means that the function is undefined at these points.

Answer

Based on the analysis above, the correct answers are:

• A. $x=-2$
• B. $x=1$

The other options, C. $x=-1$ and D. $x=10$, are not correct because they do not make the denominator of the function equal to zero.

Discussion

The concept of vertical asymptotes is an important one in mathematics, particularly in the study of rational functions. By understanding how to find vertical asymptotes, we can gain a deeper understanding of the behavior of rational functions and how they can be used to model real-world phenomena.

Real-World Applications

Vertical asymptotes have many real-world applications, particularly in the fields of physics and engineering. For example, the concept of vertical asymptotes can be used to model the behavior of electrical circuits, where the function's graph represents the current flowing through the circuit. By understanding how to find vertical asymptotes, engineers can design more efficient and effective electrical circuits.

Conclusion

Introduction

In our previous article, we explored the concept of vertical asymptotes and how to find them in rational functions. In this article, we will answer some frequently asked questions about vertical asymptotes to help you better understand this important mathematical concept.

Q: What is a vertical asymptote?

A: A vertical asymptote is a vertical line that a function approaches but never touches. It is a point where the function's graph becomes infinite or undefined.

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 set the denominator equal to zero and solve for the variable. This is because a rational function is undefined when the denominator is equal to zero.

Q: What are the values of x for which the graph of the function F(x) = x / (x+2)(x-1) has a vertical asymptote?

A: The values of x for which the graph of the function F(x) = x / (x+2)(x-1) has a vertical asymptote are x = -2 and x = 1. These values of x make the denominator of the function equal to zero, which means that the function is undefined at these points.

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 has multiple factors that are 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 hole at a point where it has a vertical asymptote?

A: Yes, a rational function can have a hole at a point where it has a vertical asymptote. This occurs when the numerator and denominator of the function have a common factor that cancels out, resulting in a hole in the graph.

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

A: To find the vertical asymptotes of a rational function with a quadratic denominator, you need to factor the denominator and set each factor equal to zero. This will give you the values of x for which the function has a vertical asymptote.

Q: Can a rational function have a vertical asymptote at x = 0?

A: Yes, a rational function can have a vertical asymptote at x = 0. This occurs when the denominator of the function is equal to zero at x = 0.

Conclusion

In conclusion, vertical asymptotes are an important concept in mathematics, particularly in the study of rational functions. By understanding how to find vertical asymptotes, you can gain a deeper understanding of the behavior of rational functions and how they can be used to model real-world phenomena. We hope that this Q&A guide has helped you better understand the concept of vertical asymptotes and how to find them in rational functions.

Additional Resources

• Khan Academy: Vertical Asymptotes
• Mathway: Vertical Asymptotes
• Wolfram Alpha: Vertical Asymptotes

Practice Problems

1. Find the vertical asymptotes of the function F(x) = x / (x+1)(x-2).
2. Determine if the function F(x) = x / (x-1)(x-2) has a vertical asymptote at x = 1.
3. Find the vertical asymptotes of the function F(x) = x / (x^2 + 1).
4. Determine if the function F(x) = x / (x+1)(x-1) has a hole at x = 1.
5. Find the vertical asymptotes of the function F(x) = x / (x^2 - 4).

Answer Key

1. x = -1 and x = 2
2. Yes
3. No vertical asymptotes
4. Yes
5. x = 2 and x = -2