Let's analyze the given functions to determine which could be the inverse of function [tex]\( g \)[/tex]:
The original function [tex]\( g \)[/tex] is given by:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & -8 & -4 & 0 & 4 & 8 \\
\hline
g(x) & 2 & 3 & 4 & 5 & 6 \\
\hline
\end{array}
\][/tex]
To find the inverse, we need to swap the roles of the [tex]\( x \)[/tex] and [tex]\( g(x) \)[/tex] values. The inverse function [tex]\( g^{-1} \)[/tex] should satisfy [tex]\( g(g^{-1}(x)) = x \)[/tex] and [tex]\( g^{-1}(g(x)) = x \)[/tex].
Looking at the possible answers, let's evaluate each:
### Option A:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & -8 & -4 & 0 & 4 & 8 \\
\hline
j(x) & -2 & -3 & -4 & 5 & 6 \\
\hline
\end{array}
\][/tex]
[tex]\( j \)[/tex] does not use the same [tex]\( x \)[/tex] values that are outputs of [tex]\( g(x) \)[/tex]. Thus, it cannot be the inverse.
### Option B:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 2 & 3 & 4 & 5 & 6 \\
\hline
k(x) & -8 & -4 & 0 & 4 & 8 \\
\hline
\end{array}
\][/tex]
Here, the [tex]\( x \)[/tex] values (2, 3, 4, 5, 6) correspond directly to the [tex]\( g(x) \)[/tex] values in the original function, and the [tex]\( k(x) \)[/tex] values map perfectly back to the original [tex]\( x \)[/tex] values. Therefore, [tex]\( k \)[/tex] is indeed the inverse of [tex]\( g \)[/tex].
### Option C:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 2 & 3 & 0 & 4 & 8 \\
\hline
m(x) & 8 & 4 & -4 & -5 & -6 \\
\hline
\end{array}
\][/tex]
[tex]\( m \)[/tex] does not correctly map [tex]\( g(x) \)[/tex] values back to the original [tex]\( x \)[/tex] values. Thus, [tex]\( m \)[/tex] cannot be the inverse.
### Option D:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & -8 & -4 & 0 & 4 & 8 \\
\hline
h(x) & -2 & -3 & -4 & -5 & -6 \\
\hline
\end{array}
\][/tex]
[tex]\( h \)[/tex] also does not use the same [tex]\( x \)[/tex] values that are outputs of [tex]\( g(x) \)[/tex]. Thus, it cannot be the inverse.
After evaluating all the options, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
