Complete the input-output table for the function [tex]$y=3^x$[/tex].

\begin{tabular}{|r|r|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
-2 & [tex]$\frac{1}{9}$[/tex] \\
\hline
-1 & [tex]$\frac{1}{3}$[/tex] \\
\hline
0 & 1 \\
\hline
1 & 3 \\
\hline
2 & 9 \\
\hline
3 & [tex]$a$[/tex] \\
\hline
4 & [tex]$b$[/tex] \\
\hline
\end{tabular}

[tex]$a = \square \quad b = \square$[/tex]



Answer :

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]