Which Statements Accurately Describe The Function $f(x)=3(16)^{\frac{4}{4} X}$? Select Three Options.A. The Initial Value Is 3.B. The Domain Is $x \ \textgreater \ 0$.C. The Range Is $y \ \textgreater \ 0$.D. The

Understanding the Function $f(x)=3(16)^{\frac{4}{4} x}$

The given function is $f(x)=3(16)^{\frac{4}{4} x}$. To determine the accuracy of the provided statements, we need to analyze the function and its properties.

Initial Value and Domain

The initial value of a function is the value of the function when the input is zero. In this case, we can find the initial value by substituting $x=0$ into the function.

$f(0)=3(16)^{\frac{4}{4} \cdot 0}$ $f(0)=3(16)^0$ $f(0)=3 \cdot 1$ $f(0)=3$

Therefore, the initial value of the function is indeed 3, making statement A accurate.

Next, let's analyze the domain of the function. The domain of a function is the set of all possible input values for which the function is defined. In this case, the function is defined for all real numbers, as there are no restrictions on the input value $x$. However, we need to consider the exponent $\frac{4}{4} x$, which can be simplified to $x$. Since the base $16$ is positive, the function will always be positive for any real value of $x$. Therefore, the domain of the function is all real numbers, and statement B is not accurate.

Range

The range of a function is the set of all possible output values for which the function is defined. In this case, the function is defined for all real numbers, and the output value is always positive, as the base $16$ is positive and the exponent $x$ is also positive. Therefore, the range of the function is all positive real numbers, making statement C accurate.

Conclusion

In conclusion, the statements that accurately describe the function $f(x)=3(16)^{\frac{4}{4} x}$ are:

• A. The initial value is 3.
• C. The range is $y \ \textgreater \ 0$.

The other statements are not accurate. Statement B is incorrect because the domain of the function is all real numbers, not just positive real numbers. Statement D is not provided, so we cannot determine its accuracy.

Final Answer

The final answer is:

• A. The initial value is 3.
• C. The range is $y \ \textgreater \ 0$.

Note: The other options are not correct.
Q&A: Understanding the Function $f(x)=3(16)^{\frac{4}{4} x}$

In the previous article, we analyzed the function $f(x)=3(16)^{\frac{4}{4} x}$ and determined that the initial value is 3 and the range is all positive real numbers. In this article, we will answer some frequently asked questions about the function.

Q: What is the simplified form of the function?

A: The simplified form of the function is $f(x)=3(16)^x$. This is because the exponent $\frac{4}{4} x$ can be simplified to $x$.

Q: Is the function an exponential function?

A: Yes, the function is an exponential function. This is because it has the form $f(x)=ab^x$, where $a$ is a constant and $b$ is the base.

Q: What is the base of the exponential function?

A: The base of the exponential function is 16.

Q: Is the function always positive?

A: Yes, the function is always positive. This is because the base 16 is positive and the exponent $x$ is also positive.

Q: What is the domain of the function?

A: The domain of the function is all real numbers. This is because there are no restrictions on the input value $x$.

Q: What is the range of the function?

A: The range of the function is all positive real numbers. This is because the function is always positive and the output value is always positive.

Q: How do I graph the function?

A: To graph the function, you can use a graphing calculator or a computer program. You can also use a table of values to plot the function.

Q: What is the horizontal asymptote of the function?

A: The horizontal asymptote of the function is $y=0$. This is because as $x$ approaches negative infinity, the function approaches 0.

Q: What is the vertical asymptote of the function?

A: There is no vertical asymptote of the function. This is because the function is defined for all real numbers.

Q: How do I find the inverse of the function?

A: To find the inverse of the function, you can swap the $x$ and $y$ values and solve for $y$. This will give you the inverse function.

Q: What is the derivative of the function?

A: To find the derivative of the function, you can use the power rule of differentiation. This will give you the derivative of the function.

Q: How do I use the function in real-world applications?

A: The function can be used in real-world applications such as modeling population growth, chemical reactions, and financial investments.

Conclusion

In conclusion, the function $f(x)=3(16)^{\frac{4}{4} x}$ is an exponential function with a base of 16 and a range of all positive real numbers. The function is always positive and has a domain of all real numbers. The function can be graphed using a graphing calculator or a computer program, and its inverse can be found by swapping the $x$ and $y$ values and solving for $y$. The function has many real-world applications and can be used to model population growth, chemical reactions, and financial investments.

Final Answer

The final answer is:

• The function $f(x)=3(16)^{\frac{4}{4} x}$ is an exponential function with a base of 16 and a range of all positive real numbers.
• The function is always positive and has a domain of all real numbers.
• The function can be graphed using a graphing calculator or a computer program, and its inverse can be found by swapping the $x$ and $y$ values and solving for $y$.
• The function has many real-world applications and can be used to model population growth, chemical reactions, and financial investments.