Vertical Asymptotes in the Given Function

Oct 13, 2025 by windailys.com 6 views

When analyzing a function, it's essential to consider various aspects, including its asymptotes, intercepts, and domain. In this article, we will delve into the function ${ f(x) = \frac{3}{x-2} }$ and explore its properties to determine which of the given descriptions accurately describe its behavior.

Asymptotes: A Key Concept in Function Analysis

An asymptote is a line that a function approaches as the input (or independent variable) gets arbitrarily close to a certain value. In the context of rational functions like ${ f(x) = \frac{3}{x-2} }$ , vertical asymptotes occur when the denominator is equal to zero, causing the function to become undefined.

Vertical Asymptotes in the Given Function

To identify the vertical asymptote(s) of the function ${ f(x) = \frac{3}{x-2} }$ , we need to find the value of ${ x }$ that makes the denominator equal to zero. In this case, the denominator is ${ x - 2 }$ , which equals zero when ${ x = 2 }$ . Therefore, the function has a vertical asymptote at ${ x = 2 }$ .

Horizontal Asymptotes: A Different Story

Horizontal asymptotes, on the other hand, occur when the function approaches a constant value as the input gets arbitrarily large. In the case of rational functions, the horizontal asymptote can be determined by comparing the degrees of the numerator and denominator.

For the function ${ f(x) = \frac{3}{x-2} }$ , the degree of the numerator is 0 (since it's a constant), and the degree of the denominator is 1. Since the degree of the denominator is higher than the degree of the numerator, the function has a horizontal asymptote at ${ y = 0 }$ .

Intercepts: Where the Function Meets the Axes

Intercepts are the points where the function intersects the x-axis (x-intercepts) or the y-axis (y-intercept). To find the x-intercept(s) of the function, we need to set the function equal to zero and solve for ${ x }$ .

For the function ${ f(x) = \frac{3}{x-2} }$ , setting it equal to zero gives us:

${ \frac{3}{x-2} = 0 }$

Since the numerator is a constant, the only way for the function to be equal to zero is if the denominator is equal to zero, which we already know is not possible. Therefore, the function has no x-intercepts.

Domain and Range: Understanding the Function's Behavior

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In the case of the function ${ f(x) = \frac{3}{x-2} }$ , the domain is all real numbers except ${ x = 2 }$ , since the function is undefined at that point.

The range of a function is the set of all possible output values (y-values) for which the function is defined. Since the function has a horizontal asymptote at ${ y = 0 }$ , the range of the function is all real numbers except zero.

Conclusion: Which of the Descriptions is Correct?

Based on our analysis, we can conclude that the function ${ f(x) = \frac{3}{x-2} }$ has:

1 vertical asymptote at ${ x = 2 }$
1 horizontal asymptote at ${ y = 0 }$
No x-intercepts
A domain of all real numbers except ${ x = 2 }$
A range of all real numbers except zero

Therefore, the correct descriptions of the function are:

A. ${ f(x) }$ has 1 asymptote B. ${ f(x) }$ has no intercepts C. ${ f(x) }$ has a domain of all real numbers except ${ x = 2 }$

In our previous article, we explored the function ${ f(x) = \frac{3}{x-2} }$ and analyzed its properties, including its asymptotes, intercepts, and domain. In this article, we will address some common questions and provide additional insights to help you better understand the function.

Q: What is the purpose of analyzing a function's asymptotes?

A: Analyzing a function's asymptotes is crucial in understanding its behavior, especially when dealing with rational functions like ${ f(x) = \frac{3}{x-2} }$ . Asymptotes help us identify the function's limits, which can be useful in various applications, such as optimization problems, physics, and engineering.

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 compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at ${ y = 0 }$ . If the degrees are equal, the horizontal asymptote is at 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 difference between a vertical asymptote and a hole in a graph?

A: A vertical asymptote occurs when the function approaches infinity or negative infinity as the input gets arbitrarily close to a certain value. A hole in a graph, on the other hand, occurs when there is a removable discontinuity, which means that the function is undefined at a specific point but can be made continuous by removing that point.

Q: How do I find the x-intercept(s) of a rational function?

A: To find the x-intercept(s) of a rational function, you need to set the function equal to zero and solve for ${ x }$ . However, if the function has a vertical asymptote, it may not have any x-intercepts. In the case of the function ${ f(x) = \frac{3}{x-2} }$ , we found that it has no x-intercepts.

Q: What is the significance of the domain and range of a function?

A: The domain and range of a function are essential in understanding its behavior and limitations. The domain represents the set of all possible input values, while the range represents the set of all possible output values. In the case of the function ${ f(x) = \frac{3}{x-2} }$ , the domain is all real numbers except ${ x = 2 }$ , and the range is all real numbers except zero.

Q: Can a function have multiple asymptotes?

A: Yes, a function can have multiple asymptotes. For example, a rational function with multiple vertical asymptotes or a function with both vertical and horizontal asymptotes. In the case of the function ${ f(x) = \frac{3}{x-2} }$ , we found that it has one vertical asymptote and one horizontal asymptote.

Q: How do I graph a rational function?

A: Graphing a rational function involves plotting the function's asymptotes, intercepts, and other key points. You can use a graphing calculator or software to visualize the function's behavior. In the case of the function ${ f(x) = \frac{3}{x-2} }$ , you can plot the vertical asymptote at ${ x = 2 }$ , the horizontal asymptote at ${ y = 0 }$ , and the function's behavior between these asymptotes.

Conclusion: Mastering the Function and Its Properties

In this article, we addressed some common questions and provided additional insights to help you better understand the function ${ f(x) = \frac{3}{x-2} }$ and its properties. By analyzing the function's asymptotes, intercepts, domain, and range, you can gain a deeper understanding of its behavior and limitations.