What Is The Horizontal Asymptote Of The Function $f(x)=\frac{(x-2)}{(x-3)^2}$?A. $y=0$ B. \$y=1$[/tex\] C. $y=2$ D. $y=3$
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 essential concept in calculus and algebra, and understanding it is crucial for analyzing and graphing functions. In this article, we will explore the concept of horizontal asymptotes and apply it to the function f(x) = (x-2)/(x-3)^2 to determine its horizontal asymptote.
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 positive or negative infinity.
Types of Horizontal Asymptotes
There are three types of horizontal asymptotes:
- Horizontal Asymptote at y = 0: If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
- Horizontal Asymptote at y = c: If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is y = c, where c is the ratio of the leading coefficients.
- No Horizontal Asymptote: If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
Analyzing the Function f(x) = (x-2)/(x-3)^2
To determine the horizontal asymptote of the function f(x) = (x-2)/(x-3)^2, we need to analyze its degree and leading coefficients.
- Degree of the Numerator: The degree of the numerator is 1, which is the highest power of x in the numerator.
- Degree of the Denominator: The degree of the denominator is 2, which is the highest power of x in the denominator.
- Leading Coefficients: The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 1.
Determining the Horizontal Asymptote
Based on the analysis above, we can conclude that the degree of the numerator is less than the degree of the denominator. Therefore, the horizontal asymptote of the function f(x) = (x-2)/(x-3)^2 is y = 0.
Conclusion
In conclusion, the horizontal asymptote of the function f(x) = (x-2)/(x-3)^2 is y = 0. This is because the degree of the numerator is less than the degree of the denominator, which means that the function approaches the x-axis as x approaches positive or negative infinity.
Final Answer
The final answer is A. y = 0.
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 an essential concept in calculus and algebra, and understanding it is crucial for analyzing and graphing functions.
Q: What are the types of horizontal asymptotes?
A: There are three types of horizontal asymptotes:
- Horizontal Asymptote at y = 0: If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
- Horizontal Asymptote at y = c: If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is y = c, where c is the ratio of the leading coefficients.
- No Horizontal Asymptote: If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
Q: How do I determine the horizontal asymptote of a function?
A: To determine the horizontal asymptote of a function, you need to analyze its degree and leading coefficients. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is y = c, where c is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
Q: What is the significance of horizontal asymptotes?
A: Horizontal asymptotes are significant in calculus and algebra because they help us understand the behavior of functions as x approaches positive or negative infinity. They are also used to determine the limits of functions and to graph functions.
Q: Can a function have more than one horizontal asymptote?
A: No, a function cannot have more than one horizontal asymptote. A horizontal asymptote is a unique line that a function approaches as x approaches positive or negative infinity.
Q: How do I graph a function with a horizontal asymptote?
A: To graph a function with a horizontal asymptote, you need to draw a horizontal line at the y-coordinate of the horizontal asymptote. Then, you need to graph the function and see how it approaches the horizontal asymptote as x approaches positive or negative infinity.
Q: What is the difference between a horizontal asymptote and a slant asymptote?
A: A horizontal asymptote is a horizontal line that a function approaches as x approaches positive or negative infinity. A slant asymptote is a line that a function approaches as x approaches positive or negative infinity, but it is not horizontal. Slant asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator.
Q: Can a function have a horizontal asymptote and a slant asymptote at the same time?
A: No, a function cannot have a horizontal asymptote and a slant asymptote at the same time. If a function has a horizontal asymptote, it means that the degree of the numerator is less than or equal to the degree of the denominator. If a function has a slant asymptote, it means that the degree of the numerator is exactly one more than the degree of the denominator.
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 analyze its degree and leading coefficients. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is y = c, where c is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
Q: Can a function have a horizontal asymptote at y = 0 and also have a slant asymptote?
A: No, a function cannot have a horizontal asymptote at y = 0 and also have a slant asymptote. If a function has a horizontal asymptote at y = 0, it means that the degree of the numerator is less than the degree of the denominator. If a function has a slant asymptote, it means that the degree of the numerator is exactly one more than the degree of the denominator.
Q: How do I determine the horizontal asymptote of a function with a hole?
A: To determine the horizontal asymptote of a function with a hole, you need to analyze the function and determine if the hole is a removable discontinuity or a non-removable discontinuity. If the hole is a removable discontinuity, you can remove it and determine the horizontal asymptote of the resulting function. If the hole is a non-removable discontinuity, you cannot remove it and the function does not have a horizontal asymptote.
Q: Can a function have a horizontal asymptote and also have a removable discontinuity?
A: Yes, a function can have a horizontal asymptote and also have a removable discontinuity. In this case, you can remove the removable discontinuity and determine the horizontal asymptote of the resulting function.
Q: How do I determine the horizontal asymptote of a function with a vertical asymptote?
A: To determine the horizontal asymptote of a function with a vertical asymptote, you need to analyze the function and determine if the vertical asymptote is a removable discontinuity or a non-removable discontinuity. If the vertical asymptote is a removable discontinuity, you can remove it and determine the horizontal asymptote of the resulting function. If the vertical asymptote is a non-removable discontinuity, you cannot remove it and the function does not have a horizontal asymptote.
Q: Can a function have a horizontal asymptote and also have a vertical asymptote?
A: Yes, a function can have a horizontal asymptote and also have a vertical asymptote. In this case, you can determine the horizontal asymptote of the function and also analyze the vertical asymptote to determine its behavior.
Q: How do I determine the horizontal asymptote of a function with a slant asymptote?
A: To determine the horizontal asymptote of a function with a slant asymptote, you need to analyze the function and determine if the slant asymptote is a removable discontinuity or a non-removable discontinuity. If the slant asymptote is a removable discontinuity, you can remove it and determine the horizontal asymptote of the resulting function. If the slant asymptote is a non-removable discontinuity, you cannot remove it and the function does not have a horizontal asymptote.
Q: Can a function have a horizontal asymptote and also have a slant asymptote?
A: No, a function cannot have a horizontal asymptote and also have a slant asymptote. If a function has a horizontal asymptote, it means that the degree of the numerator is less than or equal to the degree of the denominator. If a function has a slant asymptote, it means that the degree of the numerator is exactly one more than the degree of the denominator.
Q: How do I determine the horizontal asymptote of a function with a rational expression?
A: To determine the horizontal asymptote of a function with a rational expression, you need to analyze the degree and leading coefficients of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is y = c, where c is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
Q: Can a function have a horizontal asymptote and also have a rational expression?
A: Yes, a function can have a horizontal asymptote and also have a rational expression. In this case, you can determine the horizontal asymptote of the function and also analyze the rational expression to determine its behavior.
Q: How do I determine the horizontal asymptote of a function with a polynomial expression?
A: To determine the horizontal asymptote of a function with a polynomial expression, you need to analyze the degree and leading coefficients of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is y = c, where c is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
Q: Can a function have a horizontal asymptote and also have a polynomial expression?
A: Yes, a function can have a horizontal asymptote and also have a polynomial expression. In this case, you can determine the horizontal asymptote of the function and also analyze the polynomial expression to determine its behavior.
Q: How do I determine the horizontal asymptote of a function with a trigonometric expression?
A: To determine the horizontal asymptote of a function with a trigonometric expression, you need to analyze the function and determine if it has a horizontal asymptote. If the function has a horizontal asymptote, you can determine its value by analyzing the trigonometric expression.
Q: Can a function have a horizontal asymptote and also have a trigonometric expression?
A: Yes, a function can have a horizontal asymptote and also have a trig