Select The Correct Answer.Which Statement Describes The End Behavior Of The Function? F ( X ) = X 2 − 100 X 2 − 3 X − 4 F(x)=\frac{x^2-100}{x^2-3x-4} F ( X ) = X 2 − 3 X − 4 X 2 − 100 ​ A. The Function Approaches 0 As X X X Approaches − ∞ -\infty − ∞ And ∞ \infty ∞ . B. The Function

Introduction

When analyzing the behavior of rational functions, it is essential to consider the end behavior, which refers to the behavior of the function as $x$ approaches positive or negative infinity. In this article, we will delve into the end behavior of the function $f(x)=\frac{x^2-100}{x^2-3x-4}$ and determine which statement accurately describes its behavior.

Understanding Rational Functions

Rational functions are functions that can be expressed as the ratio of two polynomials. The general form of a rational function is $f(x)=\frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials. Rational functions can have various types of behavior, including asymptotic behavior, which is characterized by the function approaching a specific value or infinity as $x$ approaches a particular point.

End Behavior of Rational Functions

The end behavior of a rational function is determined by the degrees of the polynomials in the numerator and denominator. If the degree of the numerator is greater than the degree of the denominator, the function will have a slant asymptote. If the degree of the numerator is equal to the degree of the denominator, the function will have a horizontal asymptote. If the degree of the numerator is less than the degree of the denominator, the function will approach 0 as $x$ approaches positive or negative infinity.

Analyzing the Function

Let's analyze the function $f(x)=\frac{x^2-100}{x^2-3x-4}$. The numerator and denominator are both quadratic polynomials, so the degree of the numerator is equal to the degree of the denominator. To determine the end behavior of the function, we need to examine the leading terms of the numerator and denominator.

Leading Terms

The leading term of the numerator is $x^2$, and the leading term of the denominator is $x^2$. Since the leading terms are the same, the function will have a horizontal asymptote.

Horizontal Asymptote

To find the horizontal asymptote, we need to divide the leading term of the numerator by the leading term of the denominator. In this case, we have:

$$\frac{x^2}{x^2}=1$$

So, the horizontal asymptote is $y=1$.

End Behavior

Since the degree of the numerator is equal to the degree of the denominator, the function will approach the horizontal asymptote as $x$ approaches positive or negative infinity. Therefore, the function will approach 1 as $x$ approaches positive or negative infinity.

Conclusion

In conclusion, the end behavior of the function $f(x)=\frac{x^2-100}{x^2-3x-4}$ is that it approaches 1 as $x$ approaches positive or negative infinity. This is because the degree of the numerator is equal to the degree of the denominator, and the leading terms are the same. Therefore, the correct statement is:

The function approaches 1 as $x$ approaches $-\infty$ and $\infty$.

This statement accurately describes the end behavior of the function, and it is the correct answer.

Final Thoughts

In this article, we analyzed the end behavior of the function $f(x)=\frac{x^2-100}{x^2-3x-4}$ and determined that it approaches 1 as $x$ approaches positive or negative infinity. This is an essential concept in mathematics, and it has numerous applications in various fields, including physics, engineering, and economics. By understanding the end behavior of rational functions, we can better analyze and solve problems involving these functions.

Introduction

In our previous article, we discussed the end behavior of rational functions and analyzed the function $f(x)=\frac{x^2-100}{x^2-3x-4}$. In this article, we will address some of the most frequently asked questions related to the end behavior of rational functions.

Q: What is the end behavior of a rational function?

A: The end behavior of a rational function refers to the behavior of the function as $x$ approaches positive or negative infinity. It is an essential concept in mathematics, and it has numerous applications in various fields.

Q: How do I determine the end behavior of a rational function?

A: To determine the end behavior of a rational function, you need to examine the degrees of the polynomials in the numerator and denominator. If the degree of the numerator is greater than the degree of the denominator, the function will have a slant asymptote. If the degree of the numerator is equal to the degree of the denominator, the function will have a horizontal asymptote. If the degree of the numerator is less than the degree of the denominator, the function will approach 0 as $x$ approaches positive or negative infinity.

Q: What is a horizontal asymptote?

A: A horizontal asymptote is a horizontal line that the function approaches as $x$ approaches positive or negative infinity. It is a value that the function gets arbitrarily close to as $x$ gets arbitrarily large.

Q: How do I find the horizontal asymptote of a rational function?

A: To find the horizontal asymptote of a rational function, you need to divide the leading term of the numerator by the leading term of the denominator. This will give you the value of the horizontal asymptote.

Q: What is a slant asymptote?

A: A slant asymptote is a line that the function approaches as $x$ approaches positive or negative infinity. It is a value that the function gets arbitrarily close to as $x$ gets arbitrarily large.

Q: How do I find the slant asymptote of a rational function?

A: To find the slant asymptote of a rational function, you need to divide the numerator by the denominator using long division or synthetic division. This will give you the value of the slant asymptote.

Q: Can a rational function have both a horizontal and a slant asymptote?

A: No, a rational function cannot have both a horizontal and a slant asymptote. If the degree of the numerator is greater than the degree of the denominator, the function will have a slant asymptote. If the degree of the numerator is equal to the degree of the denominator, the function will have a horizontal asymptote.

Q: How do I determine the end behavior of a rational function with a slant asymptote?

A: To determine the end behavior of a rational function with a slant asymptote, you need to examine the degree of the numerator and the degree of the denominator. If the degree of the numerator is greater than the degree of the denominator by 1, the function will have a slant asymptote that is a linear function.

Q: Can a rational function have a vertical asymptote?

A: Yes a rational function can have a vertical asymptote. A vertical asymptote occurs when the denominator of the function is equal to 0, and the numerator is not equal to 0.

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

A: To find the vertical asymptote of a rational function, you need to set the denominator equal to 0 and solve for $x$. This will give you the value of the vertical asymptote.

Conclusion

In this article, we addressed some of the most frequently asked questions related to the end behavior of rational functions. We discussed the concept of end behavior, horizontal asymptotes, slant asymptotes, and vertical asymptotes. We also provided examples and explanations to help clarify the concepts. By understanding the end behavior of rational functions, you can better analyze and solve problems involving these functions.