Given the function [tex]\( f(x) \)[/tex] provided in the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-6 & 1 \\
\hline
-3 & 2 \\
\hline
2 & 5 \\
\hline
5 & 3 \\
\hline
8 & 0 \\
\hline
\end{array}
\][/tex]
We need to find [tex]\( f(g(2)) \)[/tex], where [tex]\( g(x) \)[/tex] is the inverse function of [tex]\( f(x) \)[/tex].
1. First, recognize that [tex]\( g(x) \)[/tex] being the inverse of [tex]\( f(x) \)[/tex] means that [tex]\( g \)[/tex] swaps the roles of the inputs and outputs of [tex]\( f \)[/tex]. In other words, if [tex]\( f(a) = b \)[/tex], then [tex]\( g(b) = a \)[/tex].
2. Identify [tex]\( g(2) \)[/tex]:
- Look in the table to see which [tex]\( x \)[/tex] value corresponds to [tex]\( f(x) = 2 \)[/tex].
- From the table: [tex]\( f(-3) = 2 \)[/tex].
Thus, [tex]\( g(2) = -3 \)[/tex].
3. Now, evaluate [tex]\( f(g(2)) \)[/tex]:
- Substitute [tex]\( g(2) \)[/tex] into [tex]\( f \)[/tex], which is [tex]\( f(-3) \)[/tex].
- From the table, we know that [tex]\( f(-3) = 2 \)[/tex].
Therefore, the value of [tex]\( f(g(2)) \)[/tex] is [tex]\( 2 \)[/tex].
[tex]\[
\boxed{2}
\][/tex]
So the correct answer is 2.
