Complete The Table Of Values.$\[ \begin{tabular}{|c|c|} \hline $x$ & $4^{-x}$ \\ \hline -1 & 4 \\ \hline 0 & $a$ \\ \hline 2 & $b$ \\ \hline 4 & $c$ \\ \hline \end{tabular} \\]$\[ \begin{array}{l} a = \square \\ b = \square \\ c =

Introduction

In mathematics, tables of values are a crucial tool for understanding and visualizing the behavior of functions. By examining the values of a function at specific points, we can gain insights into its properties, such as its domain, range, and behavior as the input variable changes. In this article, we will explore the table of values for the function $4^{-x}$ and complete the missing entries.

The Table of Values

The given table of values is as follows:

Understanding the Function

The function $4^{-x}$ is an exponential function with base 4. The negative exponent indicates that the function is decreasing as the input variable $x$ increases. To understand the behavior of this function, let's examine its properties.

• Domain: The domain of the function $4^{-x}$ is all real numbers, since the base 4 is positive and the exponent $-x$ is also real.
• Range: The range of the function $4^{-x}$ is all positive real numbers, since the base 4 is positive and the exponent $-x$ is also real.
• Behavior: As the input variable $x$ increases, the function $4^{-x}$ decreases. This is because the negative exponent causes the function to approach 0 as $x$ approaches infinity.

Completing the Table of Values

To complete the table of values, we need to find the values of $a$, $b$, and $c$.

• Value of $a$: Since $x = 0$, we have $4^{-0} = 4^0 = 1$. Therefore, the value of $a$ is 1.
• Value of $b$: Since $x = 2$, we have $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$. Therefore, the value of $b$ is $\frac{1}{16}$.
• Value of $c$: Since $x = 4$, we have $4^{-4} = \frac{1}{4^4} = \frac{1}{256}$. Therefore, the value of $c$ is $\frac{1}{256}$.

The Completed Table of Values

The completed table of values is as follows:

Conclusion

In this article, we explored the table of values for the function $4^{-x}$ and completed the missing entries. By understanding the properties of the function, we were able to determine the values of $a$, $b$, and $c$. The completed table of values provides a clear picture of the behavior of the function and can be used to gain insights into its properties.

Discussion

The table of values for the function $4^{-x}$ is a useful tool for understanding the behavior of exponential functions. By examining the values of the function at specific points, we can gain insights into its properties, such as its domain, range, and behavior as the input variable changes. The completed table of values can be used to solve problems and make predictions about the behavior of the function.

Applications

The table of values for the function $4^{-x}$ has several applications in mathematics and other fields. For example:

• Mathematics: The table of values can be used to solve problems involving exponential functions, such as finding the value of a function at a specific point or determining the behavior of a function as the input variable changes.
• Science: The table of values can be used to model real-world phenomena, such as population growth or radioactive decay.
• Engineering: The table of values can be used to design and optimize systems, such as electronic circuits or mechanical systems.

Future Directions

The table of values for the function $4^{-x}$ is a fundamental concept in mathematics and has many applications in other fields. Future research directions include:

• Exploring other exponential functions: Investigating the properties and behavior of other exponential functions, such as $2^x$ or $3^x$.
• Developing new applications: Finding new applications for the table of values in mathematics and other fields.
• Improving computational methods: Developing more efficient computational methods for calculating the values of exponential functions.
  

Introduction

In our previous article, we explored the table of values for the function $4^{-x}$ and completed the missing entries. In this article, we will answer some of the most frequently asked questions about completing the table of values.

Q: What is the domain of the function $4^{-x}$?

A: The domain of the function $4^{-x}$ is all real numbers, since the base 4 is positive and the exponent $-x$ is also real.

Q: What is the range of the function $4^{-x}$?

A: The range of the function $4^{-x}$ is all positive real numbers, since the base 4 is positive and the exponent $-x$ is also real.

Q: How do I complete the table of values for the function $4^{-x}$?

A: To complete the table of values, you need to find the values of $a$, $b$, and $c$. You can do this by substituting the values of $x$ into the function $4^{-x}$ and calculating the corresponding values.

Q: What is the value of $a$ in the table of values?

A: The value of $a$ is 1, since $4^{-0} = 4^0 = 1$.

Q: What is the value of $b$ in the table of values?

A: The value of $b$ is $\frac{1}{16}$, since $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$.

Q: What is the value of $c$ in the table of values?

A: The value of $c$ is $\frac{1}{256}$, since $4^{-4} = \frac{1}{4^4} = \frac{1}{256}$.

Q: How do I use the table of values to solve problems?

A: You can use the table of values to solve problems by substituting the values of $x$ into the function $4^{-x}$ and calculating the corresponding values. You can also use the table of values to make predictions about the behavior of the function.

Q: What are some applications of the table of values in mathematics and other fields?

A: The table of values has several applications in mathematics and other fields, including:

• Mathematics: The table of values can be used to solve problems involving exponential functions, such as finding the value of a function at a specific point or determining the behavior of a function as the input variable changes.
• Science: The table of values can be used to model real-world phenomena, such as population growth or radioactive decay.
• Engineering: The table of values can be used to design and optimize systems, such as electronic circuits or mechanical systems.

Q: What are some future directions for research on the table of values?

A: Some future directions for research on the table of values include:

• Exploring other exponential functions: Investigating the properties and behavior of other exponential functions, such as $2^x$ or $3^x$.
• Developing new applications: Finding new applications for the table of values in mathematics and other fields.
• Improving computational methods: Developing more efficient computational methods for calculating the values of exponential functions.

Conclusion

In this article, we answered some of the most frequently asked questions about completing the table of values. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of the table of values.