To solve this question, we need to create the inverse table for the function [tex]\( f \)[/tex]. In an inverse function, the roles of [tex]\( x \)[/tex] and [tex]\( f(x) \)[/tex] are reversed. Here’s a step-by-step solution:
1. Original Table:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline
x & -2 & -1 & 0 & 1 & 2 \\
\hline
f(x) & -28 & -9 & -2 & -1 & 0 \\
\hline
\end{tabular}
\][/tex]
2. Establish the Relationship:
Each [tex]\( (x, f(x)) \)[/tex] pair in the original function table helps in forming the [tex]\( (f(x), x) \)[/tex] pair for the inverse table.
3. Construct the Inverse Table:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline
x & -28 & -9 & -2 & -1 & 0 \\
\hline
f^{-1}(x) & -2 & -1 & 0 & 1 & 2 \\
\hline
\end{tabular}
\][/tex]
4. Mapping to the Given Format:
The table given in the problem is partially filled, and we need to match our inverse table to this format. According to the inverse table:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline
x & - & - & - & -1 & 0 \\
\hline
f^{-1}(x) & -2 & - & 0 & - & - \\
\hline
\end{tabular}
\][/tex]
5. Fill the Gaps:
Based on the above inverse table:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline
x & -28 & -9 & -2 & -1 & 0 \\
\hline
f^{-1}(x) & -2 & -1 & 0 & 1 & 2 \\
\hline
\end{tabular}
\][/tex]
Therefore, the correct answer filled in the given boxes will be:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & -28 & -9 & -2 & -1 & 0 \\
\hline
f^{-1}(x) & -2 & -1 & 0 & 1 & 2 \\
\hline
\end{array}
\][/tex]
So the final table modeling the inverse function is:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline
x & -28 & -9 & -2 & -1 & 0 \\
\hline
f^{-1}(x) & -2 & -1 & 0 & 1 & 2 \\
\hline
\end{tabular}
\][/tex]
