Given $f(x)=\frac{1}{x-7}+9$, Determine The Horizontal Asymptote.Horizontal Asymptote: $y =$

Introduction

In mathematics, a horizontal asymptote is a horizontal line that a function approaches as the absolute value of the x-coordinate gets larger and larger. It is an important concept in calculus and is used to determine the behavior of a function as it approaches infinity or negative infinity. In this article, we will determine the horizontal asymptote of the given rational function $f(x)=\frac{1}{x-7}+9$.

What is a Horizontal Asymptote?

A horizontal asymptote is a horizontal line that a function approaches as the absolute value of the x-coordinate gets larger and larger. It is denoted by the equation $y = c$, where $c$ is a constant. In other words, a horizontal asymptote is a line that the function gets arbitrarily close to as $x$ approaches infinity or negative infinity.

Horizontal Asymptote of a Rational Function

To determine the horizontal asymptote of a rational function, we need to examine the behavior of the function as $x$ approaches infinity or negative infinity. A rational function is a function that can be written in the form $f(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials.

Step 1: Determine the Degree of the Numerator and Denominator

The degree of a polynomial is the highest power of the variable in the polynomial. To determine the horizontal asymptote of a rational function, we need to determine the degree of the numerator and denominator.

For the given rational function $f(x)=\frac{1}{x-7}+9$, the numerator is a constant polynomial of degree 0, and the denominator is a polynomial of degree 1.

Step 2: Compare the Degrees of the Numerator and Denominator

If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote is $y = 0$. If the degree of the numerator is equal to the degree of the denominator, then the horizontal asymptote is $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$. If the degree of the numerator is greater than the degree of the denominator, then there is no horizontal asymptote.

Step 3: Determine the Horizontal Asymptote

For the given rational function $f(x)=\frac{1}{x-7}+9$, the degree of the numerator is 0, and the degree of the denominator is 1. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.

Conclusion

In this article, we determined the horizontal asymptote of the given rational function $f(x)=\frac{1}{x-7}+9$. We used the concept of horizontal asymptotes and the behavior of rational functions as $x$ approaches infinity or negative infinity. We found that the horizontal asymptote is $y = 0$.

Example Problems

1. Determine the horizontal asymptote of the rational function $f(x)=\frac{2x}{x^2+1}$.
2. Determine the horizontal asymptote of the rational function $f(x)=\frac{x^2}{x^2-1}$.

Solutions

1. The degree of the numerator is 1, and the degree of the denominator is 2. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.
2. The degree of the numerator is 2, and the degree of the denominator is 2. Since the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is $y = \frac{1}{1} = 1$.

Final Answer

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Introduction

In our previous article, we discussed the concept of horizontal asymptotes and how to determine them for rational functions. In this article, we will answer some frequently asked questions about horizontal asymptotes of rational functions.

Q: What is a horizontal asymptote?

A: A horizontal asymptote is a horizontal line that a function approaches as the absolute value of the x-coordinate gets larger and larger. It is denoted by the equation $y = c$, where $c$ is a constant.

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

A: To determine the horizontal asymptote of a rational function, you need to examine the behavior of the function as $x$ approaches infinity or negative infinity. You can do this by comparing the degrees of the numerator and denominator.

Q: What if the degree of the numerator is less than the degree of the denominator?

A: If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote is $y = 0$.

Q: What if the degree of the numerator is equal to the degree of the denominator?

A: If the degree of the numerator is equal to the degree of the denominator, then the horizontal asymptote is $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$.

Q: What if the degree of the numerator is greater than the degree of the denominator?

A: If the degree of the numerator is greater than the degree of the denominator, then there is no horizontal asymptote.

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

A: No, a rational function can only have one horizontal asymptote.

Q: Can a rational function have a horizontal asymptote of the form $y = ax + b$?

A: No, a rational function can only have a horizontal asymptote of the form $y = c$, where $c$ is a constant.

Q: How do I determine the horizontal asymptote of a rational function with a quadratic numerator and a linear denominator?

A: To determine the horizontal asymptote of a rational function with a quadratic numerator and a linear denominator, you need to compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote is $y = 0$. If the degree of the numerator is equal to the degree of the denominator, then the horizontal asymptote is $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$.

Q: How do I determine the horizontal asymptote of a rational function with a linear numerator and a quadratic denominator?

A: To determine the horizontal asymptote of a rational function with a linear numerator and a quadratic denominator, you need to compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote is $y = 0$. If the degree of the numerator is equal to the degree of the denominator, then the horizontal asymptote is $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$.

Q: Can a rational function have a horizontal asymptote of the form $y = \frac{1}{x}$?

A: No, a rational function can only have a horizontal asymptote of the form $y = c$, where $c$ is a constant.

Conclusion

In this article, we answered some frequently asked questions about horizontal asymptotes of rational functions. We discussed how to determine the horizontal asymptote of a rational function and provided examples to illustrate the concepts.

Example Problems

1. Determine the horizontal asymptote of the rational function $f(x)=\frac{2x}{x^2+1}$.
2. Determine the horizontal asymptote of the rational function $f(x)=\frac{x^2}{x^2-1}$.

Solutions

1. The degree of the numerator is 1, and the degree of the denominator is 2. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.
2. The degree of the numerator is 2, and the degree of the denominator is 2. Since the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is $y = \frac{1}{1} = 1$.

Final Answer

The horizontal asymptote of a rational function is determined by comparing the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote is $y = 0$. If the degree of the numerator is equal to the degree of the denominator, then the horizontal asymptote is $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$.