Given the table for the function [tex]\( g(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x & g(x) \\
\hline
-6 & 8 \\
\hline
-3 & 2 \\
\hline
0 & 0 \\
\hline
3 & 2 \\
\hline
6 & 8 \\
\hline
\end{array}
\][/tex]
To determine the value of [tex]\( k \)[/tex], let's analyze the values provided and identify any discernible patterns in the function [tex]\( g(x) \)[/tex].
We observe the following pairs:
- [tex]\( g(-6) = 8 \)[/tex]
- [tex]\( g(-3) = 2 \)[/tex]
- [tex]\( g(0) = 0 \)[/tex]
- [tex]\( g(3) = 2 \)[/tex]
- [tex]\( g(6) = 8 \)[/tex]
Notice the symmetry in the function values around [tex]\( x = 0 \)[/tex]:
- [tex]\( g(6) = g(-6) = 8 \)[/tex]
- [tex]\( g(3) = g(-3) = 2 \)[/tex]
- [tex]\( g(0) = 0 \)[/tex]
The symmetry indicates that [tex]\( g(x) \)[/tex] is an even function, where [tex]\( g(x) = g(-x) \)[/tex].
Given that the function values are symmetric and [tex]\( g(x) \)[/tex] does not depend on [tex]\( x \)[/tex] itself, but rather on its absolute value, we can deduce the values of [tex]\( g(x) \)[/tex] for given [tex]\( x \)[/tex] directly from the table.
Therefore, the value of [tex]\( k \)[/tex] that corresponds to this relationship is:
[tex]\[
k = \{-6: 8, -3: 2, 0: 0, 3: 2, 6: 8\}
\][/tex]
Thus, based on the function values and the observed pattern, the table accurately captures the relationship.
