To determine the initial value of the exponential function represented by the given table, we need to look at the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex]. This is because the initial value of a function typically refers to the value it takes at [tex]\( x = 0 \)[/tex].
However, in this context, "initial value" could also be interpreted as the value of [tex]\( f(x) \)[/tex] at the smallest value of [tex]\( x \)[/tex] in the table, which is [tex]\( x = -2 \)[/tex].
Looking at the table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $f(x)$ \\
\hline
-2 & $\frac{1}{8}$ \\
\hline
-1 & $\frac{1}{4}$ \\
\hline
0 & $\frac{1}{2}$ \\
\hline
1 & 1 \\
\hline
2 & 2 \\
\hline
\end{tabular}
\][/tex]
Here, when [tex]\( x = -2 \)[/tex], [tex]\( f(x) = \frac{1}{8} \)[/tex].
Thus, the initial value of the exponential function according to this table is [tex]\( \frac{1}{8} \)[/tex].
So, the correct answer is [tex]\(\frac{1}{8}\)[/tex].
