To calculate \( f^{-1}(g(2)) \), we need to follow these steps:
1. Determine the value of \( g(2) \):
From the table given:
[tex]\[
g(2) = 2
\][/tex]
2. Find \( f^{-1}(2) \):
\( f^{-1}(2) \) means we need to find the value of \( x \) such that \( f(x) = 2 \). We look at the table of \( f(x) \):
[tex]\[
\begin{array}{|l|l|l|l|l|l|l|}
\hline
x & 1 & 2 & 3 & 4 & K & Q \\
\hline
f(x) & 12 & 3 & 1 & 2 & 4 & 7 \\
\hline
\end{array}
\][/tex]
From the table, we see that \( f(4) = 2 \).
Hence, \( f^{-1}(2) = 4 \).
Therefore, \( f^{-1}(g(2)) = 4 \). The final result is:
[tex]\[ f^{-1}(g(2)) = 4 \][/tex]
