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

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

$x$	$4^{-x}$
-1	4
0	$a$
2	$b$
4	$c$

Understanding the Function

The function $4^{-x}$ is an exponential function with base 4. To understand its behavior, let's recall the properties of exponential functions. An exponential function with base $b$ and exponent $x$ is defined as $b^x$. In this case, the base is 4, and the exponent is $-x$. This means that as $x$ increases, the value of $4^{-x}$ decreases, and vice versa.

Completing the Table of Values

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

Finding the Value of $a$

To find the value of $a$, we need to substitute $x = 0$ into the function $4^{-x}$. This gives us:

$$a = 4^{-0}$$

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

$$a = 4^0 = 1$$

Finding the Value of $b$

To find the value of $b$, we need to substitute $x = 2$ into the function $4^{-x}$. This gives us:

$$b = 4^{-2}$$

Using the property of negative exponents, we can rewrite this as:

$$b = \frac{1}{4^2} = \frac{1}{16}$$

Finding the Value of $c$

To find the value of $c$, we need to substitute $x = 4$ into the function $4^{-x}$. This gives us:

$$c = 4^{-4}$$

Using the property of negative exponents, we can rewrite this as:

$$c = \frac{1}{4^4} = \frac{1}{256}$$

The Completed Table of Values

Now that we have found the values of $a$, $b$, and $c$, we can complete the table of values.

$x$	$4^{-x}$
-1	4
0	1
2	$\frac{1}{16}$
4	$\frac{1}{256}$

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 exponential functions and using the properties of negative exponents, we were able to find the values of $a$, $b$, and $c$. The completed table of values provides a visual representation of the behavior of the function and can be used to gain insights into its properties.

Discussion

The table of values for the function $4^{-x}$ can be used to explore various mathematical concepts, such as:

• Domain and Range: The table of values shows that the domain of the function is all real numbers, and the range is all positive real numbers.
• Exponential Growth and Decay: The table of values demonstrates the exponential growth and decay of the function as the input variable changes.
• Negative Exponents: The table of values illustrates the property of negative exponents, which states that $b^{-x} = \frac{1}{b^x}$.

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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 frequently asked questions about the table of values and the function $4^{-x}$.

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

A: The domain of the function $4^{-x}$ is all real numbers. This means that the function is defined for any value of $x$.

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

A: The range of the function $4^{-x}$ is all positive real numbers. This means that the function will always produce a positive value, no matter what value of $x$ is input.

Q: How does the function $4^{-x}$ behave as $x$ increases?

A: As $x$ increases, the value of $4^{-x}$ decreases. This is because the exponent $-x$ becomes more negative, which causes the function to decrease.

Q: How does the function $4^{-x}$ behave as $x$ decreases?

A: As $x$ decreases, the value of $4^{-x}$ increases. This is because the exponent $-x$ becomes less negative, which causes the function to increase.

Q: What is the value of $4^{-0}$?

A: The value of $4^{-0}$ is 1. This is because any non-zero number raised to the power of 0 is equal to 1.

Q: What is the value of $4^{-2}$?

A: The value of $4^{-2}$ is $\frac{1}{16}$. This is because $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$.

Q: What is the value of $4^{-4}$?

A: The value of $4^{-4}$ is $\frac{1}{256}$. This is because $4^{-4} = \frac{1}{4^4} = \frac{1}{256}$.

Q: How can I use the table of values to explore mathematical concepts?

A: The table of values can be used to explore various mathematical concepts, such as domain and range, exponential growth and decay, and negative exponents. By examining the values of the function at specific points, you can gain insights into its properties and behavior.

Conclusion

In this article, we answered some frequently asked questions about the table of values for the function $4^{-x}$. We hope that this Q&A article has provided you with a better understanding of the function and its properties. If you have any further questions, please don't hesitate to ask.

Additional Resources

• Table of Values: A table of values for the function $4^{-x}$.
• Exponential Functions: A tutorial on exponential functions, including their properties and behavior.
• Negative Exponents: A tutorial on negative exponents, including their properties and behavior.

Discussion

The table of values for the function $4^{-x}$ can be used to explore various mathematical concepts, such as:

• Domain and Range: The table of values shows that the domain of the function is all real numbers, and the range is all positive real numbers.
• Exponential Growth and Decay: The table of values demonstrates the exponential growth and decay of the function as the input variable changes.
• Negative Exponents: The table of values illustrates the property of negative exponents, which states that $b^{-x} = \frac{1}{b^x}$.

By exploring the table of values for the function $4^{-x}$, you can gain a deeper understanding of the mathematical concepts that underlie it.