To determine which value can replace the question mark so that the relation remains a function, we need to ensure that each [tex]\( x \)[/tex]-value corresponds to only one [tex]\( y \)[/tex]-value. Let's analyze the given set of ordered pairs:
[tex]\[
\begin{tabular}{|r|c|}
\hline$x$ & $y$ \\
\hline-4 & -3 \\
\hline-1 & 0 \\
\hline 2 & 1 \\
\hline$?$ & 4 \\
\hline 5 & 6 \\
\hline
\end{tabular}
\][/tex]
We know that for a relation to be a function, each [tex]\( x \)[/tex] must map to exactly one [tex]\( y \)[/tex]. Therefore, we need to choose an [tex]\( x \)[/tex]-value that is not already in the existing set of [tex]\( x \)[/tex]-values [tex]\([-4, -1, 2, 5]\)[/tex].
Given the options: [tex]\(-4\)[/tex], [tex]\(4\)[/tex], and [tex]\(2\)[/tex]:
- The value [tex]\(-4\)[/tex] is already an [tex]\( x \)[/tex]-value in the table.
- The value [tex]\(2\)[/tex] is also already an [tex]\( x \)[/tex]-value in the table.
This leaves [tex]\(4\)[/tex] as the only [tex]\( x \)[/tex]-value not present in the original list. Therefore, [tex]\(4\)[/tex] can be used as the [tex]\( x \)[/tex]-value for the pair [tex]\((4, 4)\)[/tex] so that the relation remains a function.
By choosing [tex]\(4\)[/tex], each [tex]\( x \)[/tex]-value in the table will be unique, ensuring that the relation remains a function.
So, the value that can replace the question mark is:
[tex]\[
\boxed{4}
\][/tex]
