To determine which table represents an exponential function, we need to identify whether one of the sets of data shows a consistent multiplicative relationship between successive \( f(x) \) values.
Let's analyze each table step-by-step.
### Table 1
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & 1 \\
\hline
1 & 3 \\
\hline
2 & 5 \\
\hline
3 & 8 \\
\hline
4 & 11 \\
\hline
\end{array}
\][/tex]
We check the ratios between successive \( f(x) \) values:
[tex]\[
\frac{3}{1} = 3, \quad \frac{5}{3} \approx 1.67, \quad \frac{8}{5} = 1.6, \quad \frac{11}{8} \approx 1.375
\][/tex]
The ratios are not consistent; hence, Table 1 does not represent an exponential function.
### Table 2
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & 1 \\
\hline
1 & 4 \\
\hline
2 & 16 \\
\hline
3 & 64 \\
\hline
4 & 256 \\
\hline
\end{array}
\][/tex]
We check the ratios between successive \( f(x) \) values:
[tex]\[
\frac{4}{1} = 4, \quad \frac{16}{4} = 4, \quad \frac{64}{16} = 4, \quad \frac{256}{64} = 4
\][/tex]
The ratios are consistently 4, which indicates a multiplicative relationship. Thus, Table 2 represents an exponential function.
### Table 3
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & 2 \\
\hline
1 & 4 \\
\hline
2 & 6 \\
\hline
3 & 10 \\
\hline
4 & 12 \\
\hline
\end{array}
\][/tex]
We check the ratios between successive \( f(x) \) values:
[tex]\[
\frac{4}{2} = 2, \quad \frac{6}{4} = 1.5, \quad \frac{10}{6} \approx 1.67, \quad \frac{12}{10} = 1.2
\][/tex]
The ratios are not consistent; hence, Table 3 does not represent an exponential function.
### Conclusion
Out of the three tables, Table 2 is the only one where the \( f(x) \) values follow a consistent multiplicative relationship:
[tex]\[
1, 4, 16, 64, 256
\][/tex]
with a common ratio of 4.
Therefore, Table 2 represents an exponential function.
