Increasing or Decreasing Nature

Oct 13, 2025 by talynt.org 127 views

Introduction

When analyzing the graph of a function, it's essential to understand its behavior, including its asymptotes, increasing or decreasing nature, and any other notable characteristics. In this article, we'll delve into the graph of the function $f(x) = \log_2 x$ and determine which statement is true about it.

The Function $f(x) = \log_2 x$

The function $f(x) = \log_2 x$ is a logarithmic function with base 2. This function is defined for all positive real numbers $x$ . The graph of this function is a curve that increases as $x$ increases, but at a decreasing rate.

Asymptotes of the Graph

An asymptote is a line that the graph of a function approaches as $x$ approaches a certain value. In the case of the function $f(x) = \log_2 x$ , the graph has a vertical asymptote at $x = 0$ . This means that as $x$ approaches 0 from the right, the value of $f(x)$ approaches negative infinity.

Increasing or Decreasing Nature

The graph of the function $f(x) = \log_2 x$ is increasing as $x$ increases. This is because the logarithmic function is a monotonically increasing function, meaning that it always increases as its input increases.

Statement A: The Graph Has an Asymptote of $y=0$ and is Decreasing as $x$ Approaches Positive Infinity

Statement A claims that the graph of the function $f(x) = \log_2 x$ has an asymptote of $y=0$ and is decreasing as $x$ approaches positive infinity. However, this statement is incorrect. The graph of the function $f(x) = \log_2 x$ has a vertical asymptote at $x = 0$ , not $y = 0$ . Additionally, the graph is increasing as $x$ increases, not decreasing.

Conclusion

In conclusion, the graph of the function $f(x) = \log_2 x$ has a vertical asymptote at $x = 0$ and is increasing as $x$ increases. Therefore, statement A is incorrect.

Key Takeaways

The graph of the function $f(x) = \log_2 x$ has a vertical asymptote at $x = 0$ .
The graph of the function $f(x) = \log_2 x$ is increasing as $x$ increases.
Statement A is incorrect.

Introduction

In our previous article, we discussed the graph of the function $f(x) = \log_2 x$ and determined that statement A is incorrect. In this article, we'll answer some frequently asked questions about the graph of this function.

Q: What is the vertical asymptote of the graph of the function $f(x) = \log_2 x$ ?

A: The vertical asymptote of the graph of the function $f(x) = \log_2 x$ is $x = 0$ . This means that as $x$ approaches 0 from the right, the value of $f(x)$ approaches negative infinity.

Q: Is the graph of the function $f(x) = \log_2 x$ increasing or decreasing?

A: The graph of the function $f(x) = \log_2 x$ is increasing as $x$ increases. This is because the logarithmic function is a monotonically increasing function, meaning that it always increases as its input increases.

Q: What is the domain of the function $f(x) = \log_2 x$ ?

A: The domain of the function $f(x) = \log_2 x$ is all positive real numbers $x$ . This means that $x$ must be greater than 0 for the function to be defined.

Q: What is the range of the function $f(x) = \log_2 x$ ?

A: The range of the function $f(x) = \log_2 x$ is all real numbers $y$ . This means that the function can take on any real value as $x$ increases.

Q: How does the graph of the function $f(x) = \log_2 x$ compare to the graph of the function $f(x) = \log x$ ?

A: The graph of the function $f(x) = \log_2 x$ is steeper than the graph of the function $f(x) = \log x$ . This is because the base 2 logarithm grows faster than the natural logarithm.

Q: Can the graph of the function $f(x) = \log_2 x$ be transformed into the graph of the function $f(x) = \log x$ ?

A: Yes, the graph of the function $f(x) = \log_2 x$ can be transformed into the graph of the function $f(x) = \log x$ by replacing $x$ with $2^x$ . This will stretch the graph horizontally and compress it vertically.

Conclusion

In conclusion, the graph of the function $f(x) = \log_2 x$ has a vertical asymptote at $x = 0$ and is increasing as $x$ increases. We've also answered some frequently asked questions about the graph of this function.

Key Takeaways

The graph of the function $f(x) = \log_2 x$ has a vertical asymptote at $x = 0$ .
The graph of the function $f(x) = \log_2 x$ is increasing as $x$ increases.
The domain of the function $f(x) = \log_2 x$ is all positive real numbers $x$ .
The range of the function $f(x) = \log_2 x$ is all real numbers $y$ .