Which Statements Accurately Describe The Function F ( X ) = 3 ( 18 ) X F(x) = 3(\sqrt{18})^x F ( X ) = 3 ( 18 ​ ) X ? Select Three Options.A. The Domain Is All Real Numbers.B. The Range Is Y \textgreater 3 Y \ \textgreater \ 3 Y \textgreater 3 .C. The Initial Value Is 3.D. The Initial Value Is 9.E. The

The given function $f(x) = 3(\sqrt{18})^x$ is an exponential function that involves a square root. To understand its behavior and properties, we need to analyze its components and how they interact with each other.

Domain and Range Analysis

The domain of a function refers to the set of all possible input values for which the function is defined. In the case of the given function, the expression $(\sqrt{18})^x$ is defined for all real values of $x$. However, since the square root of a negative number is not a real number, we need to consider the properties of the square root function.

The square root of 18 is a positive real number, and raising it to any power will result in a positive real number. Therefore, the domain of the function $f(x) = 3(\sqrt{18})^x$ is all real numbers.

On the other hand, the range of a function refers to the set of all possible output values. In this case, the function is an exponential function with a positive base, which means that it will always produce positive output values. However, the function is multiplied by a constant factor of 3, which means that the output values will be scaled by a factor of 3.

Therefore, the range of the function $f(x) = 3(\sqrt{18})^x$ is all positive real numbers greater than 3.

Initial Value Analysis

The initial value of a function refers to the value of the function at a specific input value, usually $x = 0$. In this case, we need to find the value of the function at $x = 0$.

To do this, we can substitute $x = 0$ into the function:

$f(0) = 3(\sqrt{18})^0$

Since any number raised to the power of 0 is equal to 1, we have:

$f(0) = 3(1) = 3$

Therefore, the initial value of the function $f(x) = 3(\sqrt{18})^x$ is 3.

Conclusion

In conclusion, the function $f(x) = 3(\sqrt{18})^x$ has the following properties:

• The domain is all real numbers.
• The range is all positive real numbers greater than 3.
• The initial value is 3.

Therefore, the correct statements that accurately describe the function are:

A. The domain is all real numbers. C. The initial value is 3.

The other options are incorrect:

B. The range is $y \ \textgreater \ 3$ is incorrect because the range is all positive real numbers greater than 3, not just $y \ \textgreater \ 3$. D. The initial value is 9 is incorrect because the initial value is 3, not 9. E. This option is not provided.

Final Answer

The final answer is:

In the previous article, we analyzed the function $f(x) = 3(\sqrt{18})^x$ and determined its domain, range, and initial value. However, we received several questions from readers who wanted to know more about the function. In this article, we will answer some of the most frequently asked questions about the function.

Q: What is the significance of the square root in the function?

A: The square root in the function $f(x) = 3(\sqrt{18})^x$ is used to create a positive real number that is raised to a power. The square root of 18 is a positive real number, and raising it to any power will result in a positive real number. This is important because it allows the function to produce positive output values.

Q: Can the function be simplified?

A: Yes, the function $f(x) = 3(\sqrt{18})^x$ can be simplified. Since $\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$, we can rewrite the function as:

$f(x) = 3(3\sqrt{2})^x$

This simplification makes it easier to analyze the function and understand its behavior.

Q: What happens to the function as x approaches infinity?

A: As $x$ approaches infinity, the function $f(x) = 3(\sqrt{18})^x$ will also approach infinity. This is because the exponential function will continue to grow without bound as $x$ increases.

Q: Can the function be used to model real-world phenomena?

A: Yes, the function $f(x) = 3(\sqrt{18})^x$ can be used to model real-world phenomena that involve exponential growth. For example, the function could be used to model the growth of a population, the spread of a disease, or the growth of a financial investment.

Q: How can the function be used in calculus?

A: The function $f(x) = 3(\sqrt{18})^x$ can be used in calculus to study the behavior of exponential functions. For example, the function could be used to find the derivative of an exponential function, or to study the convergence of a series.

Q: Can the function be used to solve equations?

A: Yes, the function $f(x) = 3(\sqrt{18})^x$ can be used to solve equations that involve exponential functions. For example, the function could be used to solve an equation of the form $3(\sqrt{18})^x = 5$, where $x$ is the unknown variable.

Q: What are some common applications of the function?

A: The function $f(x) = 3(\sqrt{18})^x$ has several common applications in mathematics and science. Some of these applications include:

• Modeling population growth
• Studying the spread of diseases
• Analyzing financial investments
• Solving equations involving exponential functions
• Studying the behavior of exponential functions in calculus

Conclusion

In conclusion, the function $f(x) = 3(\sqrt{18})^x$ is a powerful tool that can be used to model real-world phenomena and solve equations involving exponential functions. Its applications are numerous and varied, and it is an important concept in mathematics and science.

Final Answer

The final answer is:

• The function $f(x) = 3(\sqrt{18})^x$ is a powerful tool that can be used to model real-world phenomena and solve equations involving exponential functions.
• The function has several common applications in mathematics and science, including modeling population growth, studying the spread of diseases, and analyzing financial investments.
• The function can be used to solve equations involving exponential functions, and it is an important concept in calculus.