To complete the input-output table for the function [tex]\( y = 3^x \)[/tex], let's follow the function step-by-step for the given values of [tex]\( x \)[/tex]:
Given the function [tex]\( y = 3^x \)[/tex]:
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[
y = 3^{-2} = \frac{1}{3^2} = \frac{1}{9}
\][/tex]
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[
y = 3^{-1} = \frac{1}{3^1} = \frac{1}{3}
\][/tex]
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 3^0 = 1
\][/tex]
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 3^1 = 3
\][/tex]
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[
y = 3^2 = 9
\][/tex]
6. For [tex]\( x = 3 \)[/tex]:
[tex]\[
y = 3^3 = 27
\][/tex]
7. For [tex]\( x = 4 \)[/tex]:
[tex]\[
y = 3^4 = 81
\][/tex]
As a result, we now have the values for [tex]\( y \)[/tex] when [tex]\( x = 3 \)[/tex] and [tex]\( x = 4 \)[/tex]. Thus, [tex]\( a = 27 \)[/tex] and [tex]\( b = 81 \)[/tex]:
[tex]\[
\begin{array}{|r|r|}
\hline
x & y \\
\hline
-2 & \frac{1}{9} \\
\hline
-1 & \frac{1}{3} \\
\hline
0 & 1 \\
\hline
1 & 3 \\
\hline
2 & 9 \\
\hline
3 & 27 \\
\hline
4 & 81 \\
\hline
\end{array}
\][/tex]
Therefore:
[tex]\[
a = 27 \quad \text{and} \quad b = 81
\][/tex]
