Suppose there are two functions [tex]f[/tex] and [tex]g[/tex], whose values are defined by the table below. Calculate [tex]f^{-1}(g(2))[/tex].

\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
[tex]$x$[/tex] & 1 & 2 & 3 & 4 & [tex]$K$[/tex] & [tex]$Q$[/tex] \\
\hline
[tex]$f(x)$[/tex] & 12 & 3 & 1 & 2 & 4 & 7 \\
\hline
[tex]$g(x)$[/tex] & 11 & 2 & 4 & 1 & 8 & 7 \\
\hline
\end{tabular}



Answer :

Let's tackle this problem step-by-step.

1. Identify \( g(2) \):
- We need to find the value of the function \( g \) at \( x = 2 \).
- According to the table, \( g(2) = 2 \).

2. Find \( f^{-1}(g(2)) \):
- Now, we need to determine the value of \( f^{-1}(y) \) where \( y = g(2) \). This means we need to find \( x \) such that \( f(x) = 2 \).

3. Look up \( f(x) = 2 \) in the table:
- From the table, the value of \( f(x) = 2 \) occurs when \( x = 4 \).

Therefore, \( f^{-1}(g(2)) = 4 \).

So the detailed steps give us the final result:
[tex]\[ f^{-1}(g(2)) = 4 \][/tex]