As x Approaches Negative Infinity

Oct 13, 2025 by miodeya.com 101 views

Introduction

When analyzing the behavior of a function, particularly its end behavior, it is essential to understand how the function behaves as the input values approach positive or negative infinity. In this article, we will explore the end behavior of the function $f(x) = \log(x-2)$ and determine which statement accurately describes this behavior.

The Function

The given function is $f(x) = \log(x-2)$ . This is a logarithmic function with a base that is not explicitly stated, but it can be assumed to be the natural logarithm, denoted as $\ln$ . The function is defined only for values of $x$ greater than 2, since the argument of a logarithmic function must be positive.

End Behavior

To determine the end behavior of the function, we need to consider what happens as the input values approach positive or negative infinity. In this case, we are interested in the behavior as $x$ approaches infinity.

As x Approaches Infinity

As $x$ approaches infinity, the value of $x-2$ also approaches infinity. Since the logarithm of a number that approaches infinity also approaches infinity, we can conclude that as $x$ approaches infinity, $f(x)$ approaches negative infinity.

As x Approaches Negative Infinity

However, the function $f(x) = \log(x-2)$ is not defined for values of $x$ less than 2. Therefore, we cannot consider the behavior of the function as $x$ approaches negative infinity.

Analyzing the Statements

Now that we have a clear understanding of the end behavior of the function, let's analyze the given statements.

Statement A

As the value of $x$ decreases, the value of $f(x)$ moves toward positive infinity.

This statement is incorrect because, as we discussed earlier, the function is not defined for values of $x$ less than 2. Therefore, it is not possible for the value of $f(x)$ to move toward positive infinity as the value of $x$ decreases.

Statement B

As the value of $x$ increases, the value of $f(x)$ moves toward negative infinity.

This statement is correct because, as we discussed earlier, as $x$ approaches infinity, the value of $f(x)$ approaches negative infinity.

Conclusion

In conclusion, the end behavior of the function $f(x) = \log(x-2)$ is that as $x$ approaches infinity, $f(x)$ approaches negative infinity. Therefore, the correct statement is:

As the value of $x$ increases, the value of $f(x)$ moves toward negative infinity.

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

Additional Considerations

It is worth noting that the end behavior of a function can be affected by various factors, such as the presence of asymptotes or holes in the graph. In this case, the function has a vertical asymptote at $x=2$ , which affects its behavior as $x$ approaches 2 from the left or right.

Final Thoughts

In conclusion, understanding the end behavior of a function is crucial in analyzing its behavior and making predictions about its behavior as the input values approach positive or negative infinity. By carefully examining the function and its properties, we can determine the correct statement that describes its end behavior.

References

[1] "Calculus" by Michael Spivak
[2] "Calculus: Early Transcendentals" by James Stewart

Glossary

End behavior: The behavior of a function as the input values approach positive or negative infinity.
Logarithmic function: A function that is the inverse of an exponential function.
Vertical asymptote: A vertical line that the graph of a function approaches but never touches.
Q&A: Understanding the End Behavior of a Logarithmic Function ===========================================================

Introduction

In our previous article, we explored the end behavior of the function $f(x) = \log(x-2)$ and determined that as $x$ approaches infinity, $f(x)$ approaches negative infinity. In this article, we will answer some frequently asked questions about the end behavior of logarithmic functions and provide additional insights into this topic.

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

A: The end behavior of a logarithmic function depends on the base of the logarithm and the input values. In general, as the input values approach positive or negative infinity, the logarithmic function approaches negative infinity.

Q: How does the base of the logarithm affect the end behavior?

A: The base of the logarithm affects the end behavior of the function. For example, if the base of the logarithm is greater than 1, the function will approach negative infinity as the input values approach positive infinity. If the base of the logarithm is less than 1, the function will approach positive infinity as the input values approach positive infinity.

Q: What is the relationship between logarithmic and exponential functions?

A: Logarithmic and exponential functions are inverse functions. This means that if $f(x) = \log(x)$ , then $f^{-1}(x) = e^x$ . The end behavior of a logarithmic function is related to the end behavior of its inverse exponential function.

Q: How do vertical asymptotes affect the end behavior of a logarithmic function?

A: Vertical asymptotes can affect the end behavior of a logarithmic function. If a logarithmic function has a vertical asymptote at $x=a$ , then the function will approach negative infinity as $x$ approaches $a$ from the right and positive infinity as $x$ approaches $a$ from the left.

Q: Can a logarithmic function have a horizontal asymptote?

A: No, a logarithmic function cannot have a horizontal asymptote. This is because the logarithmic function approaches negative infinity as the input values approach positive or negative infinity.

Q: How do holes in the graph affect the end behavior of a logarithmic function?

A: Holes in the graph of a logarithmic function do not affect the end behavior of the function. The end behavior of the function is determined by the behavior of the function as the input values approach positive or negative infinity.

Q: Can a logarithmic function have a periodic end behavior?

A: No, a logarithmic function cannot have a periodic end behavior. This is because the logarithmic function approaches negative infinity as the input values approach positive or negative infinity.

Q: How do logarithmic functions compare to other types of functions in terms of end behavior?

A: Logarithmic functions have a unique end behavior compared to other types of functions. For example, polynomial functions can have a variety of end behaviors, including approaching positive or negative infinity or approaching a horizontal asymptote. Exponential functions can also have a variety of end behaviors, including approaching positive or negative infinity.

Conclusion

In conclusion, understanding the end behavior of a logarithmic function is crucial in analyzing its behavior and making predictions about its behavior as the input values approach positive or negative infinity. By carefully examining the function and its properties, we can determine the correct statement that describes its end behavior.

Additional Resources

[1] "Calculus" by Michael Spivak
[2] "Calculus: Early Transcendentals" by James Stewart
[3] "Logarithmic Functions" by Math Open Reference

Glossary

End behavior: The behavior of a function as the input values approach positive or negative infinity.
Logarithmic function: A function that is the inverse of an exponential function.
Vertical asymptote: A vertical line that the graph of a function approaches but never touches.
Horizontal asymptote: A horizontal line that the graph of a function approaches as the input values approach positive or negative infinity.
Hole: A point on the graph of a function where the function is not defined but the graph approaches the point as the input values approach the point from the left or right.