To find the decay factor of the exponential function represented by the table, we need to calculate the ratios of consecutive function values ([tex]\(f(x)\)[/tex]) as [tex]\(x\)[/tex] increases by 1 unit. This is because, in an exponential function of the form [tex]\(f(x) = a \cdot b^x\)[/tex], where [tex]\(b\)[/tex] is the decay factor, the ratio of successive function values is constant.
Given the table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $f(x)$ \\
\hline
-1 & 18 \\
\hline
0 & 6 \\
\hline
1 & 2 \\
\hline
2 & \frac{2}{3} \\
\hline
\end{tabular}
\][/tex]
1. Calculate the ratio of [tex]\(f(0)\)[/tex] to [tex]\(f(-1)\)[/tex]:
[tex]\[
\frac{f(0)}{f(-1)} = \frac{6}{18} = \frac{1}{3}
\][/tex]
2. Calculate the ratio of [tex]\(f(1)\)[/tex] to [tex]\(f(0)\)[/tex]:
[tex]\[
\frac{f(1)}{f(0)} = \frac{2}{6} = \frac{1}{3}
\][/tex]
3. Calculate the ratio of [tex]\(f(2)\)[/tex] to [tex]\(f(1)\)[/tex]:
[tex]\[
\frac{f(2)}{f(1)} = \frac{2/3}{2} = \frac{2/3}{2} = \frac{1}{3}
\][/tex]
As you can see, the ratio of consecutive [tex]\(f(x)\)[/tex] values is consistently [tex]\(\frac{1}{3}\)[/tex].
Therefore, the decay factor of the exponential function represented by the table is [tex]\(\frac{1}{3}\)[/tex]. Among the answer choices, the correct one is:
[tex]\[
\boxed{\frac{1}{3}}
\][/tex]
