Complete The Input-output Table For The Function Y = 3 X Y=3^x Y = 3 X .$[ \begin{array}{|r|r|} \hline x & Y \ \hline -2 & \square \ \hline -1 & \frac{1}{3} \ \hline 0 & 1 \ \hline 1 & 3 \ \hline 2 & 9 \ \hline 3 & A \ \hline 4 & B

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Introduction

In mathematics, functions are used to describe the relationship between two variables. The function $y=3^x$ is an exponential function that represents the power of 3 raised to the value of $x$. In this article, we will complete the input-output table for the function $y=3^x$.

Understanding the Function

The function $y=3^x$ is an exponential function that can be written in the form $y=a^x$, where $a$ is the base and $x$ is the exponent. In this case, the base is 3 and the exponent is $x$. The function can be evaluated for different values of $x$ to obtain the corresponding values of $y$.

Evaluating the Function

To evaluate the function $y=3^x$, we can use the property of exponents that states $a^x = (a^m)^n = a^{m \cdot n}$, where $m$ and $n$ are integers. Using this property, we can rewrite the function as $y=3^x = (3^1)^x = 3^{1 \cdot x} = 3^x$.

Completing the Input-Output Table

The input-output table for the function $y=3^x$ is given below:

$x$	$y$
-2	$\square$
-1	$\frac{1}{3}$
0	1
1	3
2	9
3	$a$
4	$b$

To complete the table, we need to evaluate the function for $x=-2, 3, 4$.

Evaluating the Function for $x=-2$

To evaluate the function for $x=-2$, we can use the property of exponents that states $a^{-n} = \frac{1}{a^n}$, where $n$ is a positive integer. Using this property, we can rewrite the function as $y=3^{-2} = \frac{1}{3^2} = \frac{1}{9}$.

Evaluating the Function for $x=3$

To evaluate the function for $x=3$, we can use the property of exponents that states $a^{m \cdot n} = (a^m)^n$, where $m$ and $n$ are integers. Using this property, we can rewrite the function as $y=3^3 = (3^1)^3 = 3^{1 \cdot 3} = 3^3$.

Evaluating the Function for $x=4$

To evaluate the function for $x=4$, we can use the property of exponents that states $a^{m \cdot n} = (a^m)^n$, where $m$ and $n$ are integers. Using this property, we can rewrite the function as $y=3^4 = (3^1)^4 = 3^{1 \cdot 4} = 3^4$.

Completing the Table

Using the values obtained in the previous sections, we can complete the input-output table for the function $y=3^x$ as follows:

$x$	$y$
-2	$\frac{1}{9}$
-1	$\frac{1}{3}$
0	1
1	3
2	9
3	27
4	81

Conclusion

In this article, we completed the input-output table for the function $y=3^x$. We evaluated the function for different values of $x$ and obtained the corresponding values of $y$. The completed table is given above.

Discussion

The function $y=3^x$ is an exponential function that represents the power of 3 raised to the value of $x$. The input-output table for the function can be used to evaluate the function for different values of $x$. The table can be used to identify the relationship between the input and output values of the function.

Applications

The function $y=3^x$ has many applications in mathematics and science. It can be used to model population growth, chemical reactions, and other exponential processes. The function can also be used to solve problems involving exponential decay and growth.

References

• [1] "Exponential Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/exponential.html
• [2] "Exponential Growth and Decay" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f5f7f/exponential-growth-decay

Keywords

• Exponential function
• Input-output table
• Function evaluation
• Exponential growth
• Exponential decay
  

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Introduction

In our previous article, we completed the input-output table for the function $y=3^x$. In this article, we will answer some frequently asked questions about the function and its input-output table.

Q: What is the function $y=3^x$?

A: The function $y=3^x$ is an exponential function that represents the power of 3 raised to the value of $x$. It can be written in the form $y=a^x$, where $a$ is the base and $x$ is the exponent.

Q: How do I evaluate the function $y=3^x$ for different values of $x$?

A: To evaluate the function $y=3^x$ for different values of $x$, you can use the property of exponents that states $a^x = (a^m)^n = a^{m \cdot n}$, where $m$ and $n$ are integers.

Q: What is the input-output table for the function $y=3^x$?

A: The input-output table for the function $y=3^x$ is a table that shows the input values of $x$ and the corresponding output values of $y$. The table is given below:

$x$	$y$
-2	$\frac{1}{9}$
-1	$\frac{1}{3}$
0	1
1	3
2	9
3	27
4	81

Q: How do I complete the input-output table for the function $y=3^x$?

A: To complete the input-output table for the function $y=3^x$, you can evaluate the function for different values of $x$ using the property of exponents.

Q: What are some applications of the function $y=3^x$?

A: The function $y=3^x$ has many applications in mathematics and science. It can be used to model population growth, chemical reactions, and other exponential processes. The function can also be used to solve problems involving exponential decay and growth.

Q: How do I use the input-output table for the function $y=3^x$ to solve problems?

A: To use the input-output table for the function $y=3^x$ to solve problems, you can identify the relationship between the input and output values of the function. You can then use this relationship to solve problems involving exponential growth and decay.

Q: What are some common mistakes to avoid when working with the function $y=3^x$?

A: Some common mistakes to avoid when working with the function $y=3^x$ include:

• Not using the correct property of exponents to evaluate the function
• Not completing the input-output table for the function
• Not using the input-output table to solve problems involving exponential growth and decay

Q: How do I graph the function $y=3^x$?

A: To graph the function $y=3^x$, you can use a graphing calculator or a computer program. You can also use the input-output table for the function to graph the function.

Q: What are some real-world applications of the function $y=3^x$?

A: Some real-world applications of the function $y=3^x$ include:

• Modeling population growth
• Modeling chemical reactions
• Modeling other exponential processes
• Solving problems involving exponential decay and growth

Conclusion

In this article, we answered some frequently asked questions about the function $y=3^x$ and its input-output table. We hope that this article has been helpful in understanding the function and its applications.

Discussion

The function $y=3^x$ is an important concept in mathematics and science. It has many applications in modeling population growth, chemical reactions, and other exponential processes. The input-output table for the function can be used to evaluate the function for different values of $x$ and to solve problems involving exponential growth and decay.

References

• [1] "Exponential Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/exponential.html
• [2] "Exponential Growth and Decay" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f5f7f/exponential-growth-decay

Keywords

• Exponential function
• Input-output table
• Function evaluation
• Exponential growth
• Exponential decay
• Graphing functions
• Real-world applications