Answer :
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.
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.
