Understanding the Basics of Asymptotes and Increasing Functions

Oct 13, 2025 by windailys.com 114 views

Which Function is Increasing and Has a Vertical Asymptote at $x=5$ ?

Understanding the Basics of Asymptotes and Increasing Functions

In mathematics, particularly in calculus and algebra, functions are used to describe the relationship between variables. A function can be increasing, decreasing, or constant, depending on its behavior. A vertical asymptote is a vertical line that a function approaches but never touches. In this article, we will explore which function is increasing and has a vertical asymptote at $x=5$ .

What is a Vertical Asymptote?

A vertical asymptote is a vertical line that a function approaches but never touches. It is a line that the function gets arbitrarily close to but never crosses. Vertical asymptotes occur when the function is undefined at a particular point, usually due to division by zero or a logarithmic function with a non-positive argument.

What is an Increasing Function?

An increasing function is a function that always increases as the input variable increases. In other words, as the input variable $x$ increases, the output value $f(x)$ also increases. Increasing functions are often represented by a graph that slopes upward from left to right.

Analyzing the Options

Now, let's analyze each of the given options to determine which function is increasing and has a vertical asymptote at $x=5$ .

Option A: $f(x)=\ln x+5$

The function $f(x)=\ln x+5$ is a logarithmic function with a base of $e$ . The natural logarithm function $\ln x$ is increasing for all $x>0$ . Adding a constant $5$ to the function shifts the graph upward but does not affect its increasing behavior. However, this function does not have a vertical asymptote at $x=5$ because the logarithmic function is defined for all $x>0$ .

Option B: $f(x)=-\ln x+5$

The function $f(x)=-\ln x+5$ is also a logarithmic function with a base of $e$ . The negative sign in front of the logarithm function makes it decreasing for all $x>0$ . Adding a constant $5$ to the function shifts the graph upward but does not affect its decreasing behavior. This function does not have a vertical asymptote at $x=5$ because the logarithmic function is defined for all $x>0$ .

Option C: $f(x)=\ln (x-5)$

The function $f(x)=\ln (x-5)$ is a logarithmic function with a base of $e$ . The argument of the logarithm function is $x-5$ , which means that the function is undefined when $x-5\leq 0$ . This occurs when $x\leq 5$ , which includes the point $x=5$ . Therefore, this function has a vertical asymptote at $x=5$ . However, this function is decreasing for all $x>5$ because the logarithmic function is decreasing for all $x>0$ .

Option D: $f(x)=-\ln (x-5)$

The function $f(x)=-\ln (x-5)$ is also a logarithmic function with a base of $e$ . The negative sign in front of the logarithm function makes it increasing for all $x>5$ . The argument of the logarithm function is $x-5$ , which means that the function is undefined when $x-5\leq 0$ . This occurs when $x\leq 5$ , which includes the point $x=5$ . Therefore, this function has a vertical asymptote at $x=5$ . This function is increasing for all $x>5$ because the logarithmic function is increasing for all $x>0$ .

Conclusion

Based on the analysis of each option, we can conclude that the function $f(x)=-\ln (x-5)$ is the only function that is increasing and has a vertical asymptote at $x=5$ . This function is increasing for all $x>5$ and has a vertical asymptote at $x=5$ because the logarithmic function is undefined when $x-5\leq 0$ .

Final Answer

The final answer is D.
Q&A: Which Function is Increasing and Has a Vertical Asymptote at $x=5$ ?

Understanding the Basics of Asymptotes and Increasing Functions

Q: What is a Vertical Asymptote?

A: A vertical asymptote is a vertical line that a function approaches but never touches. It is a line that the function gets arbitrarily close to but never crosses. Vertical asymptotes occur when the function is undefined at a particular point, usually due to division by zero or a logarithmic function with a non-positive argument.

Q: What is an Increasing Function?

A: An increasing function is a function that always increases as the input variable increases. In other words, as the input variable $x$ increases, the output value $f(x)$ also increases. Increasing functions are often represented by a graph that slopes upward from left to right.

Q: How do I Determine if a Function is Increasing or Decreasing?

A: To determine if a function is increasing or decreasing, you can use the following methods:

Graph the function and observe its behavior.
Take the derivative of the function and check its sign. If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing.
Check the behavior of the function at different points. If the function is increasing at all points, it is an increasing function.

Q: What is the Difference between a Vertical Asymptote and a Hole?

A: A vertical asymptote is a vertical line that a function approaches but never touches. A hole, on the other hand, is a point where the function is undefined but approaches a finite value. Holes occur when there is a removable discontinuity in the function.

Q: How do I Determine if a Function has a Vertical Asymptote?

A: To determine if a function has a vertical asymptote, you can use the following methods:

Check if the function is undefined at a particular point.
Check if the function approaches a finite value as the input variable approaches the point.
Check if the function has a removable discontinuity at the point.

Q: What is the Significance of a Vertical Asymptote?

A: A vertical asymptote is significant because it indicates that the function is undefined at a particular point. This can be useful in understanding the behavior of the function and making predictions about its behavior.

Q: How do I Use Vertical Asymptotes in Real-World Applications?

A: Vertical asymptotes can be used in real-world applications such as:

Modeling population growth and decline.
Analyzing the behavior of physical systems.
Understanding the behavior of economic systems.

Conclusion

In conclusion, a vertical asymptote is a vertical line that a function approaches but never touches. An increasing function is a function that always increases as the input variable increases. By understanding the basics of asymptotes and increasing functions, you can determine which function is increasing and has a vertical asymptote at $x=5$ .

Final Answer

The final answer is D.

Additional Resources

For more information on vertical asymptotes and increasing functions, please refer to the following resources:

Calculus textbooks such as "Calculus" by Michael Spivak.
Online resources such as Khan Academy and MIT OpenCourseWare.
Mathematical software such as Mathematica and Maple.

FAQs

Q: What is the difference between a vertical asymptote and a hole? A: A vertical asymptote is a vertical line that a function approaches but never touches. A hole, on the other hand, is a point where the function is undefined but approaches a finite value.

Q: How do I determine if a function has a vertical asymptote? A: To determine if a function has a vertical asymptote, you can check if the function is undefined at a particular point, or if the function approaches a finite value as the input variable approaches the point.

Q: What is the significance of a vertical asymptote? A: A vertical asymptote is significant because it indicates that the function is undefined at a particular point. This can be useful in understanding the behavior of the function and making predictions about its behavior.