At Which Values Of $x$ Does The Function $F(x$\] Have A Vertical Asymptote? Check All That Apply.$F(x) = \frac{9}{(x-1)(x-8)}$A. -1 B. 8 C. -9 D. 1 E. -8 F. 9

Introduction

In mathematics, a vertical asymptote is a vertical line that a function approaches but never touches. It occurs when the function is undefined at a particular point, usually due to division by zero. In this article, we will explore the concept of vertical asymptotes in rational functions, specifically the function $F(x) = \frac{9}{(x-1)(x-8)}$. We will determine the values of $x$ at which this function 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 function is undefined at a particular point, usually due to division by zero. In other words, a vertical asymptote is a line that the function gets arbitrarily close to but never crosses.

Rational Functions and Vertical Asymptotes

Rational functions are functions that can be written in the form $\frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials. The function $F(x) = \frac{9}{(x-1)(x-8)}$ is a rational function, where $p(x) = 9$ and $q(x) = (x-1)(x-8)$.

Finding 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. In other words, we need to solve the equation $q(x) = 0$. For the function $F(x) = \frac{9}{(x-1)(x-8)}$, we need to solve the equation $(x-1)(x-8) = 0$.

Solving the Equation

To solve the equation $(x-1)(x-8) = 0$, we can use the zero-product property, which states that if $ab = 0$, then either $a = 0$ or $b = 0$. Applying this property to the equation, we get:

$(x-1) = 0$ or $(x-8) = 0$

Solving for $x$ in each equation, we get:

$x = 1$ or $x = 8$

Conclusion

In conclusion, the function $F(x) = \frac{9}{(x-1)(x-8)}$ has vertical asymptotes at $x = 1$ and $x = 8$. Therefore, the correct answers are:

• A. -1: Incorrect, because $x = -1$ is not a solution to the equation $(x-1)(x-8) = 0$.
• B. 8: Correct, because $x = 8$ is a solution to the equation $(x-1)(x-8) = 0$.
• C. -9: Incorrect, because $x = -9$ is not a solution to the equation $(x-1)(x-8) = 0$.
• D. 1: Correct, because $x = 1$ is a solution to the equation $(x-1)(x-8) = 0$.
• E. -8: Incorrect, because $x = -8$ is not a solution to the equation $(x-1)(x-8) = 0$.
• F. 9: Incorrect, because $x = 9$ is not a solution to the equation $(x-1)(x-8) = 0$.

Final Answer

The final answer is:

• B. 8
• D. 1

Introduction

In our previous article, we explored the concept of vertical asymptotes in rational functions, specifically the function $F(x) = \frac{9}{(x-1)(x-8)}$. We determined that the function has vertical asymptotes at $x = 1$ and $x = 8$. In this article, we will answer some frequently asked questions about vertical asymptotes in rational functions.

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 function is undefined at a particular point, usually due to division by zero.

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. In other words, you need to solve the equation $q(x) = 0$, where $q(x)$ is the denominator of the rational function.

Q: What is the zero-product property?

A: The zero-product property is a mathematical property that states that if $ab = 0$, then either $a = 0$ or $b = 0$. This property is used to solve equations of the form $(x-a)(x-b) = 0$.

Q: How do I use the zero-product property to solve equations?

A: To use the zero-product property to solve equations, you need to set each factor equal to zero and solve for $x$. For example, if you have the equation $(x-1)(x-8) = 0$, you can use the zero-product property to solve for $x$ as follows:

$(x-1) = 0$ or $(x-8) = 0$

Solving for $x$ in each equation, you get:

$x = 1$ or $x = 8$

Q: What are some common mistakes to avoid when finding vertical asymptotes?

A: Some common mistakes to avoid when finding vertical asymptotes include:

• Not factoring the denominator correctly
• Not using the zero-product property to solve equations
• Not checking for holes in the graph (i.e., points where the function is undefined but the denominator is not equal to zero)

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

A: Yes, a rational function can have more than one vertical asymptote. For example, the function $F(x) = \frac{1}{(x-1)(x-2)(x-3)}$ has vertical asymptotes at $x = 1$, $x = 2$, and $x = 3$.

Q: Can a rational function have no vertical asymptotes?

A: Yes, a rational function can have no vertical asymptotes. For example, the function $F(x) = \frac{1}{x^2+1}$ has no vertical asymptotes because the denominator is never equal to zero.

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 better analyze and graph rational functions. We hope this Q&A article has been helpful in answering some of your questions about vertical asymptotes.

Final Answer

The final answer is:

• Yes, a rational function can have more than one vertical asymptote.
• Yes, a rational function can have no vertical asymptotes.