Which Is A Feature Of Function $g$ If $g(x)=-4 \log (x-8)$?A. The Domain Is \$x \ \textless \ 8$[/tex\]. B. The Value Of The Function Decreases As $x$ Approaches Positive Infinity. C. The Value Of The

Mar 1, 2025 by ADMIN 212 views

$**Understanding the Properties of the Function $g(x)=-4 \log (x-8)$**$

Introduction

In mathematics, functions are used to describe the relationship between variables. A function can be represented in various forms, including algebraic, trigonometric, and logarithmic. In this article, we will focus on the properties of a specific function, $g(x)=-4 \log (x-8)$, and determine which of the given features is a characteristic of this function.

Domain of the Function

The domain of a function is the set of all possible input values for which the function is defined. In the case of the function $g(x)=-4 \log (x-8)$, we need to consider the restrictions on the input values.

The logarithmic function is defined only for positive real numbers. Therefore, the expression inside the logarithm, $x-8$, must be greater than zero.
This implies that $x>8$, as the minimum value of $x-8$ is zero, and the logarithm is undefined at zero.

Conclusion

Based on the analysis above, the domain of the function $g(x)=-4 \log (x-8)$ is $x>8$. This means that option A, which states that the domain is $x<8$, is incorrect.

Range of the Function

The range of a function is the set of all possible output values. In the case of the function $g(x)=-4 \log (x-8)$, we need to consider the behavior of the logarithmic function.

As $x$ approaches infinity, the value of $x-8$ also approaches infinity.
The logarithmic function increases as its input increases. Therefore, the value of $-4 \log (x-8)$ decreases as $x$ approaches infinity.

Conclusion

Based on the analysis above, the value of the function $g(x)=-4 \log (x-8)$ decreases as $x$ approaches positive infinity. This means that option B is correct.

Asymptotic Behavior

The asymptotic behavior of a function refers to its behavior as the input values approach positive or negative infinity.

As $x$ approaches infinity, the value of $x-8$ also approaches infinity.
The logarithmic function increases as its input increases. Therefore, the value of $-4 \log (x-8)$ decreases as $x$ approaches infinity.

Conclusion

Based on the analysis above, the value of the function $g(x)=-4 \log (x-8)$ decreases as $x$ approaches positive infinity. This means that option B is correct.

Comparison with Other Functions

To further understand the properties of the function $g(x)=-4 \log (x-8)$, let's compare it with other functions.

The function $f(x)=\log x$ is defined only for positive real numbers. Therefore, the domain of $f(x)$ is $x>0$.
The function $h(x)=-\log x$ is defined only for positive real numbers. Therefore, the domain of $h(x)$ is $x>0$.

Conclusion

Based on the comparison above, the function $g(x)=-4 \log (x-8)$ has a domain of $x>8$, which is different from the domains of the functions $f(x)$ and $h(x)$.

Conclusion

In conclusion, the function $g(x)=-4 \log (x-8)$ has a domain of $x>8$ and a range that decreases as $x$ approaches positive infinity. Therefore, option B is the correct feature of the function $g(x)$.

Final Answer

Introduction

In our previous article, we explored the properties of the function $g(x)=-4 \log (x-8)$. We determined that the domain of the function is $x>8$ and that the value of the function decreases as $x$ approaches positive infinity. In this article, we will answer some frequently asked questions about the function $g(x)$.

Q: What is the domain of the function $g(x)=-4 \log (x-8)$?

A: The domain of the function $g(x)=-4 \log (x-8)$ is $x>8$. This is because the logarithmic function is defined only for positive real numbers, and the expression inside the logarithm, $x-8$, must be greater than zero.

Q: Why is the domain of the function $g(x)$ different from the domains of the functions $f(x)$ and $h(x)$?

A: The domain of the function $g(x)$ is different from the domains of the functions $f(x)$ and $h(x)$ because the expression inside the logarithm, $x-8$, is shifted by 8 units to the right. This means that the function $g(x)$ is defined only for values of $x$ that are greater than 8.

Q: What happens to the value of the function $g(x)$ as $x$ approaches positive infinity?

A: As $x$ approaches positive infinity, the value of the function $g(x)$ decreases. This is because the logarithmic function increases as its input increases, and the negative sign in front of the logarithm causes the value of the function to decrease.

Q: How does the function $g(x)$ compare to other functions, such as $f(x)=\log x$ and $h(x)=-\log x$?

A: The function $g(x)$ has a domain of $x>8$, which is different from the domains of the functions $f(x)$ and $h(x)$. Additionally, the function $g(x)$ has a range that decreases as $x$ approaches positive infinity, which is different from the ranges of the functions $f(x)$ and $h(x)$.

Q: What are some real-world applications of the function $g(x)=-4 \log (x-8)$?

A: The function $g(x)=-4 \log (x-8)$ has several real-world applications, including:

Modeling population growth and decline
Analyzing economic data
Studying the behavior of complex systems

Conclusion

In conclusion, the function $g(x)=-4 \log (x-8)$ has a domain of $x>8$ and a range that decreases as $x$ approaches positive infinity. It has several real-world applications and can be compared to other functions, such as $f(x)=\log x$ and $h(x)=-\log x$. We hope that this article has helped to clarify the properties of the function $g(x)$ and its applications.

Frequently Asked Questions

What is the domain of the function $g(x)=-4 \log (x-8)$?
Why is the domain of the function $g(x)$ different from the domains of the functions $f(x)$ and $h(x)$?
What happens to the value of the function $g(x)$ as $x$ approaches positive infinity?
How does the function $g(x)$ compare to other functions, such as $f(x)=\log x$ and $h(x)=-\log x$?
What are some real-world applications of the function $g(x)=-4 \log (x-8)$?

Answers

The domain of the function $g(x)=-4 \log (x-8)$ is $x>8$.
The domain of the function $g(x)$ is different from the domains of the functions $f(x)$ and $h(x)$ because the expression inside the logarithm, $x-8$, is shifted by 8 units to the right.
As $x$ approaches positive infinity, the value of the function $g(x)$ decreases.
The function $g(x)$ has a domain of $x>8$, which is different from the domains of the functions $f(x)$ and $h(x)$. Additionally, the function $g(x)$ has a range that decreases as $x$ approaches positive infinity, which is different from the ranges of the functions $f(x)$ and $h(x)$.
The function $g(x)=-4 \log (x-8)$ has several real-world applications, including modeling population growth and decline, analyzing economic data, and studying the behavior of complex systems.