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Which table represents an exponential function?

\begin{tabular}{|c|c|}
\hline [tex]$x$[/tex] & [tex]$f(x)$[/tex] \\
\hline 0 & 1 \\
\hline 1 & 3 \\
\hline 2 & 5 \\
\hline 3 & 8 \\
\hline 4 & 11 \\
\hline
\end{tabular}

\begin{tabular}{|c|c|}
\hline [tex]$x$[/tex] & [tex]$f(x)$[/tex] \\
\hline 0 & 1 \\
\hline 1 & 4 \\
\hline 2 & 16 \\
\hline 3 & 64 \\
\hline 4 & 256 \\
\hline
\end{tabular}

\begin{tabular}{|c|c|}
\hline [tex]$x$[/tex] & [tex]$f(x)$[/tex] \\
\hline 0 & 2 \\
\hline 1 & 4 \\
\hline 2 & 6 \\
\hline 3 & 10 \\
\hline 4 & 12 \\
\hline
\end{tabular}

Sagot :

GinnyAnswer
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.
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