To determine which table represents an exponential function, let's examine the nature of exponential functions and how they manifest through given data points.
An exponential function has the form:
[tex]\[ f(x) = a \cdot b^x \][/tex]
where [tex]\( b \)[/tex] is a constant base and [tex]\( a \)[/tex] is a constant coefficient.
For each table, we'll check if the ratio of consecutive [tex]\( f(x) \)[/tex]-values remains constant, a key characteristic of exponential growth or decay.
### Table 1 Analysis:
[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]
Ratios,
[tex]\[
\frac{f(1)}{f(0)} = \frac{3}{1} = 3, \quad \frac{f(2)}{f(1)} = \frac{5}{3}, \quad \frac{f(3)}{f(2)} = \frac{8}{5}, \quad \frac{f(4)}{f(3)} = \frac{11}{8}
\][/tex]
These ratios are not equal. Therefore, Table 1 does not represent an exponential function.
### Table 2 Analysis:
[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]
Ratios,
[tex]\[
\frac{f(1)}{f(0)} = \frac{4}{1} = 4, \quad \frac{f(2)}{f(1)} = \frac{16}{4} = 4, \quad \frac{f(3)}{f(2)} = \frac{64}{16} = 4, \quad \frac{f(4)}{f(3)} = \frac{256}{64} = 4
\][/tex]
These ratios are all equal. Therefore, Table 2 represents an exponential function with a base [tex]\( b = 4 \)[/tex].
### Table 3 Analysis:
[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]
Ratios,
[tex]\[
\frac{f(1)}{f(0)} = \frac{4}{2} = 2, \quad \frac{f(2)}{f(1)} = \frac{6}{4} = 1.5, \quad \frac{f(3)}{f(2)} = \frac{10}{6} \approx 1.67, \quad \frac{f(4)}{f(3)} = \frac{12}{10} = 1.2
\][/tex]
These ratios are not equal. Therefore, Table 3 does not represent an exponential function.
### Conclusion:
Upon inspecting the ratios, it is evident that Table 2 has a consistent ratio between consecutive [tex]\( f(x) \)[/tex] values. Thus, Table 2 represents an exponential function.
