Select All The Correct Answers.Consider The Graph Of The Function F ( X ) = Log ⁡ 2 X F(x)=\log _2 X F ( X ) = Lo G 2 ​ X .What Are The Features Of The Function G G G If G ( X ) = F ( X + 4 ) + 8 G(x)=f(x+4)+8 G ( X ) = F ( X + 4 ) + 8 ?- Y Y Y -intercept At ( 0 , 10 (0,10 ( 0 , 10 ]- Domain Of ( 4 , ∞ (4, \infty ( 4 , ∞ ]-

Introduction

Logarithmic functions are a fundamental concept in mathematics, and understanding their properties and transformations is crucial for solving various mathematical problems. In this article, we will explore the transformation of the function $f(x) = \log_2 x$ to obtain the function $g(x) = f(x+4) + 8$. We will analyze the features of the function $g$, including its $y$-intercept and domain.

The Original Function

The original function is $f(x) = \log_2 x$. This function is a logarithmic function with base 2, and its domain is $(0, \infty)$. The range of the function is all real numbers.

The Transformed Function

The transformed function is $g(x) = f(x+4) + 8$. To understand this function, we need to analyze the transformations applied to the original function $f(x)$.

• Horizontal Shift: The function $f(x)$ is shifted 4 units to the left. This means that the input $x$ is replaced by $x+4$. As a result, the graph of the function $g$ is shifted 4 units to the left compared to the graph of the function $f$.
• Vertical Shift: The function $f(x+4)$ is shifted 8 units upwards. This means that the output of the function $f(x+4)$ is increased by 8. As a result, the graph of the function $g$ is shifted 8 units upwards compared to the graph of the function $f$.

Features of the Function $g$

Now that we have analyzed the transformations applied to the original function $f(x)$, we can determine the features of the function $g$.

• $y$-intercept: The $y$-intercept of the function $g$ is $(0, 10)$. This is because the function $f(x+4)$ has a $y$-intercept at $(4, \log_2 4)$, and when we shift the graph 8 units upwards, the $y$-intercept becomes $(0, \log_2 4 + 8) = (0, 10)$.
• Domain: The domain of the function $g$ is $(4, \infty)$. This is because the function $f(x+4)$ has a domain of $(4, \infty)$, and the horizontal shift does not change the domain.

Conclusion

In conclusion, the function $g(x) = f(x+4) + 8$ is a transformed version of the original function $f(x) = \log_2 x$. The function $g$ has a $y$-intercept at $(0, 10)$ and a domain of $(4, \infty)$. Understanding the properties and transformations of logarithmic functions is essential for solving various mathematical problems.

Key Takeaways

• The function $g(x) = f(x+4) + 8$ is a transformed version of the original function $f(x) = \log_2 x$.
• The function $g$ has a $y$-intercept at $(0, 10)$.
• The domain of the function $g$ is $(4, \infty)$.

Further Reading

For more information on logarithmic functions and their transformations, we recommend the following resources:

• Khan Academy: Logarithmic Functions
• Mathway: Logarithmic Functions
• Wolfram MathWorld: Logarithmic Functions

Introduction

In our previous article, we explored the transformation of the function $f(x) = \log_2 x$ to obtain the function $g(x) = f(x+4) + 8$. We analyzed the features of the function $g$, including its $y$-intercept and domain. In this article, we will answer some frequently asked questions about the transformation of logarithmic functions.

Q: What is the effect of the horizontal shift on the graph of the function $g$?

A: The horizontal shift of 4 units to the left means that the graph of the function $g$ is shifted 4 units to the left compared to the graph of the function $f$. This means that the input $x$ is replaced by $x+4$, resulting in a shift of the graph to the left.

Q: How does the vertical shift affect the graph of the function $g$?

A: The vertical shift of 8 units upwards means that the output of the function $f(x+4)$ is increased by 8. This results in a shift of the graph of the function $g$ 8 units upwards compared to the graph of the function $f$.

Q: What is the $y$-intercept of the function $g$?

A: The $y$-intercept of the function $g$ is $(0, 10)$. This is because the function $f(x+4)$ has a $y$-intercept at $(4, \log_2 4)$, and when we shift the graph 8 units upwards, the $y$-intercept becomes $(0, \log_2 4 + 8) = (0, 10)$.

Q: What is the domain of the function $g$?

A: The domain of the function $g$ is $(4, \infty)$. This is because the function $f(x+4)$ has a domain of $(4, \infty)$, and the horizontal shift does not change the domain.

Q: How can I apply the transformation to other logarithmic functions?

A: To apply the transformation to other logarithmic functions, you can follow these steps:

1. Identify the original function and its domain.
2. Determine the horizontal and vertical shifts required.
3. Apply the horizontal shift by replacing the input $x$ with $x+4$.
4. Apply the vertical shift by adding 8 to the output of the function.
5. Analyze the features of the transformed function, including its $y$-intercept and domain.

Q: What are some common applications of logarithmic functions?

A: Logarithmic functions have many applications in various fields, including:

• Mathematics: Logarithmic functions are used to solve equations and inequalities, and to model real-world phenomena.
• Science: Logarithmic functions are used to model population growth, chemical reactions, and other scientific phenomena.
• Engineering: Logarithmic functions are used to design and optimize systems, such as electronic circuits and mechanical systems.
• Finance: Logarithmic functions are used to model stock prices and other financial data.

Conclusion

In conclusion, the transformation of logarithmic functions is a powerful tool for solving mathematical problems and modeling real-world phenomena. By understanding the properties and transformations of logarithmic functions, you can apply them to various fields and gain a deeper insight into the world of mathematics.

Key Takeaways

• The function $g(x) = f(x+4) + 8$ is a transformed version of the original function $f(x) = \log_2 x$.
• The function $g$ has a $y$-intercept at $(0, 10)$.
• The domain of the function $g$ is $(4, \infty)$.
• Logarithmic functions have many applications in various fields, including mathematics, science, engineering, and finance.

Further Reading

For more information on logarithmic functions and their transformations, we recommend the following resources:

• Khan Academy: Logarithmic Functions
• Mathway: Logarithmic Functions
• Wolfram MathWorld: Logarithmic Functions

By understanding the properties and transformations of logarithmic functions, you can solve various mathematical problems and gain a deeper insight into the world of mathematics.