To determine whether a given relation represents a function, we need to check if each input (or [tex]\(x\)[/tex]-value) is associated with exactly one output (or [tex]\(y\)[/tex]-value).
Let's examine the given options:
Option B: [tex]$\{(-1,-11),(0,-7),(1,-3),(-1,5),(2,0)\}$[/tex]
In this set, we notice that the input [tex]\(-1\)[/tex] is associated with two different outputs: [tex]\(-11\)[/tex] and [tex]\(5\)[/tex]. Therefore, this relation is not a function because a single input has been paired with more than one output.
Option D:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$x$ & -18 & -13 & 3 & 5 & -6 & 3 \\
\hline
$y$ & -7 & -2 & 14 & 16 & 5 & 19 \\
\hline
\end{tabular}
\][/tex]
In this table, the input [tex]\(3\)[/tex] is associated with two different outputs: [tex]\(14\)[/tex] and [tex]\(19\)[/tex]. Therefore, this relation is not a function because a single input corresponds to multiple outputs.
Given the choices and the conditions that a function must meet, none of the options provided clearly represent a valid function. Options B and D definitely do not meet the criteria for functions as inputs are repeated with different outputs.
Since options A and C are not provided, and both options B and D do not satisfy the definition of a function, we conclude that:
None of the options listed correspond to a function.
