Answer :
To find \( g(f^{-1}(1)) \), we need to follow these steps:
1. Find \( f^{-1}(1) \):
Determine the value of \( x \) for which \( f(x) = 1 \).
Looking at the table for \( f(x) \):
[tex]\[ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 1 & 2 & 3 & 4 & K & Q \\ \hline f(x) & 12 & 3 & 1 & 2 & 4 & 7 \\ \hline \end{array} \][/tex]
We see that \( f(3) = 1 \), hence \( f^{-1}(1) = 3 \).
2. Calculate \( g(f^{-1}(1)) \):
Now that we have \( f^{-1}(1) = 3 \), we need to find \( g(3) \).
Looking at the table for \( g(x) \):
[tex]\[ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 1 & 2 & 3 & 4 & K & Q \\ \hline g(x) & 11 & 2 & 4 & 1 & 8 & 7 \\ \hline \end{array} \][/tex]
We find that \( g(3) = 4 \).
Therefore, the value of \( g(f^{-1}(1)) \) is \( 4 \).
[tex]\[ \boxed{4} \][/tex]
1. Find \( f^{-1}(1) \):
Determine the value of \( x \) for which \( f(x) = 1 \).
Looking at the table for \( f(x) \):
[tex]\[ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 1 & 2 & 3 & 4 & K & Q \\ \hline f(x) & 12 & 3 & 1 & 2 & 4 & 7 \\ \hline \end{array} \][/tex]
We see that \( f(3) = 1 \), hence \( f^{-1}(1) = 3 \).
2. Calculate \( g(f^{-1}(1)) \):
Now that we have \( f^{-1}(1) = 3 \), we need to find \( g(3) \).
Looking at the table for \( g(x) \):
[tex]\[ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 1 & 2 & 3 & 4 & K & Q \\ \hline g(x) & 11 & 2 & 4 & 1 & 8 & 7 \\ \hline \end{array} \][/tex]
We find that \( g(3) = 4 \).
Therefore, the value of \( g(f^{-1}(1)) \) is \( 4 \).
[tex]\[ \boxed{4} \][/tex]
