To determine which function represents the inverse of the given function, we need to understand the concept of inverse functions. An inverse function essentially swaps the x and f(x) values of the original function.
Given the table for the function [tex]\(f(x)\)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & -1 & 0 & 1 & 2 \\
\hline
f(x) & -2 & 3 & 8 & 13 \\
\hline
\end{array}
\][/tex]
To find the inverse function, we switch the x and f(x) values:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & -2 & 3 & 8 & 13 \\
\hline
f^{-1}(x) & -1 & 0 & 1 & 2 \\
\hline
\end{array}
\][/tex]
Now, let's compare this table to the provided options:
A.
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & -1 & 0 & 1 & 2 \\
\hline
p(x) & 2 & -3 & -8 & -13 \\
\hline
\end{array}
\][/tex]
Option A does not match our derived inverse table.
B.
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & -2 & 3 & 8 & 13 \\
\hline
q(x) & -1 & 0 & 1 & 2 \\
\hline
\end{array}
\][/tex]
Option B matches our derived inverse table exactly.
C.
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 1 & 0 & -1 & -2 \\
\hline
s(x) & -2 & 3 & 8 & 13 \\
\hline
\end{array}
\][/tex]
Option C does not match our derived inverse table.
D.
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 2 & -3 & -8 & -13 \\
\hline
r(x) & 1 & 0 & -1 & -2 \\
\hline
\end{array}
\][/tex]
Option D does not match our derived inverse table.
Based on this, the correct option that represents the inverse of the given function is Option B.
