Select The Correct Answer.Consider The Graph Of The Function F ( X ) = Log ⁡ X F(x)=\log X F ( X ) = Lo G X .Which Is A Feature Of Function G G G If G ( X ) = Log ⁡ ( X + 2 G(x)=\log (x+2 G ( X ) = Lo G ( X + 2 ]?A. Horizontal Asymptote Of Y = 2 Y=2 Y = 2 B. Vertical Asymptote Of X = − 2 X=-2 X = − 2 C. Domain

When dealing with logarithmic functions, it's essential to understand the characteristics of their graphs. The function $f(x) = \log x$ is a fundamental example of a logarithmic function. In this article, we will explore the graph of $f(x)$ and use it to determine the features of a related function $g(x) = \log (x+2)$.

The Graph of $f(x) = \log x$

The graph of $f(x) = \log x$ is a logarithmic curve that passes through the point $(1, 0)$. This curve has a vertical asymptote at $x = 0$, which means that the function approaches positive infinity as $x$ approaches 0 from the right. The graph also has a horizontal asymptote at $y = -\infty$, indicating that the function approaches negative infinity as $x$ approaches positive infinity.

The Domain of $f(x) = \log x$

The domain of $f(x) = \log x$ is all positive real numbers, denoted as $(0, \infty)$. This means that the function is defined for all values of $x$ greater than 0.

The Graph of $g(x) = \log (x+2)$

Now, let's consider the function $g(x) = \log (x+2)$. This function is related to $f(x)$, but it has a horizontal shift of 2 units to the left. In other words, the graph of $g(x)$ is the same as the graph of $f(x)$, but shifted 2 units to the left.

Features of the Graph of $g(x)$

Based on the graph of $f(x)$, we can determine the features of the graph of $g(x)$. Since the graph of $g(x)$ is a horizontal shift of 2 units to the left of the graph of $f(x)$, it will have the same vertical asymptote at $x = -2$. This means that the function $g(x)$ will approach positive infinity as $x$ approaches -2 from the right.

Horizontal Asymptote of $g(x)$

The horizontal asymptote of $g(x)$ is the same as the horizontal asymptote of $f(x)$, which is $y = -\infty$. This means that the function $g(x)$ will approach negative infinity as $x$ approaches positive infinity.

Domain of $g(x)$

The domain of $g(x)$ is all real numbers greater than -2, denoted as $(-2, \infty)$. This means that the function is defined for all values of $x$ greater than -2.

Conclusion

In conclusion, the graph of $g(x) = \log (x+2)$ is a horizontal shift of 2 units to the left of the graph of $f(x) = \log x$. Based on the graph of $f(x)$, we can determine the features of the graph of $g(x)$, including the vertical asymptote at $x = -2$, the horizontal asymptote at $y = -\infty$, and the domain of $(-2, \infty)$.

Answer

Based on the analysis above, the correct answer is:

• B. Vertical asymptote of $x=-2$

This is because the graph of $g(x)$ has a vertical asymptote at $x = -2$, which is a feature of the function $g(x) = \log (x+2)$.

Discussion

The graph of a logarithmic function is a fundamental concept in mathematics. Understanding the characteristics of the graph of a logarithmic function is essential for analyzing and solving problems involving logarithmic functions. In this article, we have explored the graph of $f(x) = \log x$ and used it to determine the features of the graph of $g(x) = \log (x+2)$. We have also discussed the domain of $g(x)$ and the horizontal asymptote of $g(x)$.

Key Takeaways

• The graph of $f(x) = \log x$ is a logarithmic curve that passes through the point $(1, 0)$.
• The graph of $f(x)$ has a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = -\infty$.
• The domain of $f(x)$ is all positive real numbers, denoted as $(0, \infty)$.
• The graph of $g(x) = \log (x+2)$ is a horizontal shift of 2 units to the left of the graph of $f(x)$.
• The graph of $g(x)$ has a vertical asymptote at $x = -2$ and a horizontal asymptote at $y = -\infty$.
• The domain of $g(x)$ is all real numbers greater than -2, denoted as $(-2, \infty)$.

References

• [1] "Logarithmic Functions" by Math Open Reference
• [2] "Graphs of Logarithmic Functions" by Purplemath
• [3] "Logarithmic Functions" by Khan Academy
  Q&A: Understanding the Graph of a Logarithmic Function =====================================================

In our previous article, we explored the graph of the function $f(x) = \log x$ and used it to determine the features of the graph of $g(x) = \log (x+2)$. In this article, we will answer some frequently asked questions about the graph of a logarithmic function.

Q: What is the domain of a logarithmic function?

A: The domain of a logarithmic function is all positive real numbers. This means that the function is defined for all values of $x$ greater than 0.

Q: What is the vertical asymptote of a logarithmic function?

A: The vertical asymptote of a logarithmic function is at $x = 0$. This means that the function approaches positive infinity as $x$ approaches 0 from the right.

Q: What is the horizontal asymptote of a logarithmic function?

A: The horizontal asymptote of a logarithmic function is at $y = -\infty$. This means that the function approaches negative infinity as $x$ approaches positive infinity.

Q: How does the graph of $g(x) = \log (x+2)$ differ from the graph of $f(x) = \log x$?

A: The graph of $g(x) = \log (x+2)$ is a horizontal shift of 2 units to the left of the graph of $f(x) = \log x$. This means that the graph of $g(x)$ is the same as the graph of $f(x)$, but shifted 2 units to the left.

Q: What is the domain of $g(x) = \log (x+2)$?

A: The domain of $g(x) = \log (x+2)$ is all real numbers greater than -2. This means that the function is defined for all values of $x$ greater than -2.

Q: What is the vertical asymptote of $g(x) = \log (x+2)$?

A: The vertical asymptote of $g(x) = \log (x+2)$ is at $x = -2$. This means that the function approaches positive infinity as $x$ approaches -2 from the right.

Q: What is the horizontal asymptote of $g(x) = \log (x+2)$?

A: The horizontal asymptote of $g(x) = \log (x+2)$ is at $y = -\infty$. This means that the function approaches negative infinity as $x$ approaches positive infinity.

Q: How can I determine the features of the graph of a logarithmic function?

A: To determine the features of the graph of a logarithmic function, you can use the following steps:

1. Determine the domain of the function.
2. Determine the vertical asymptote of the function.
3. Determine the horizontal asymptote of the function.
4. Use the graph of a related function to determine the features of the graph of the original function.

Q: What are some common mistakes to avoid when working with logarithmic functions?

A: Some common mistakes to avoid when working with logarithmic functions include:

1. Assuming that the domain of a logarithmic function is all real numbers.
2. Failing to determine the vertical asymptote of a logarithmic function.
3. Failing to determine the horizontal asymptote of a logarithmic function.
4. Not using the graph of a related function to determine the features of the graph of the original function.

Conclusion

In conclusion, understanding the graph of a logarithmic function is essential for analyzing and solving problems involving logarithmic functions. By following the steps outlined in this article, you can determine the features of the graph of a logarithmic function and avoid common mistakes.