Let's analyze each table to determine which one represents an exponential function.
### Table 1:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $f(x)$ \\
\hline
0 & 1 \\
\hline
1 & 3 \\
\hline
2 & 5 \\
\hline
3 & 8 \\
\hline
4 & 11 \\
\hline
\end{tabular}
\][/tex]
An exponential function has the form [tex]\( f(x) = a \cdot b^x \)[/tex] where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants. Let's examine the ratios of successive [tex]\( f(x) \)[/tex] values:
[tex]\[
\frac{f(1)}{f(0)} = \frac{3}{1} = 3
\][/tex]
[tex]\[
\frac{f(2)}{f(1)} = \frac{5}{3} \approx 1.67
\][/tex]
[tex]\[
\frac{f(3)}{f(2)} = \frac{8}{5} = 1.6
\][/tex]
[tex]\[
\frac{f(4)}{f(3)} = \frac{11}{8} \approx 1.375
\][/tex]
The ratios are not constant, suggesting that this table does not represent an exponential function.
### Table 2:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $f(x)$ \\
\hline
0 & 1 \\
\hline
1 & 4 \\
\hline
2 & 16 \\
\hline
3 & 64 \\
\hline
4 & 256 \\
\hline
\end{tabular}
\][/tex]
Again, we examine the ratios of successive [tex]\( f(x) \)[/tex] values:
[tex]\[
\frac{f(1)}{f(0)} = \frac{4}{1} = 4
\][/tex]
[tex]\[
\frac{f(2)}{f(1)} = \frac{16}{4} = 4
\][/tex]
[tex]\[
\frac{f(3)}{f(2)} = \frac{64}{16} = 4
\][/tex]
[tex]\[
\frac{f(4)}{f(3)} = \frac{256}{64} = 4
\][/tex]
The ratios are constant and equal to 4, suggesting that the function [tex]\( f(x) \)[/tex] in this table can be described by an exponential equation of the form [tex]\( f(x) = 1 \cdot 4^x \)[/tex]. Thus, Table 2 represents an exponential function.
### Table 3:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $f(x)$ \\
\hline
0 & 2 \\
\hline
1 & 4 \\
\hline
2 & 6 \\
\hline
3 & 10 \\
\hline
4 & 12 \\
\hline
\end{tabular}
\][/tex]
Again, we examine the ratios of successive [tex]\( f(x) \)[/tex] values:
[tex]\[
\frac{f(1)}{f(0)} = \frac{4}{2} = 2
\][/tex]
[tex]\[
\frac{f(2)}{f(1)} = \frac{6}{4} = 1.5
\][/tex]
[tex]\[
\frac{f(3)}{f(2)} = \frac{10}{6} \approx 1.67
\][/tex]
[tex]\[
\frac{f(4)}{f(3)} = \frac{12}{10} = 1.2
\][/tex]
The ratios are not constant, suggesting that this table does not represent an exponential function.
### Conclusion
Of the provided tables, only Table 2 consistently shows constant ratios between successive [tex]\( f(x) \)[/tex] values, which is characteristic of an exponential function. Therefore, Table 2 represents an exponential function.
