To determine which relation is not a function, we need to understand the definition of a function in mathematics. A relation is defined as a function if every element of the domain (the set of all possible input values, typically represented as [tex]\(x\)[/tex]) is associated with exactly one element of the range (the set of all possible output values, typically represented as [tex]\(y\)[/tex]).
In other words, for a relation to be a function, each input value must map to only one output value. If any input value maps to more than one output value, the relation is not a function.
Let's examine the relation provided in the table:
[tex]\[
\begin{tabular}{|c|c|}
\hline$x$ & $y$ \\
\hline-4 & 0 \\
\hline-3 & 9 \\
\hline 5 & 2 \\
\hline 6 & 9 \\
\hline
\end{tabular}
\][/tex]
We have pairs of [tex]\((x, y)\)[/tex] as follows:
- (-4, 0)
- (-3, 9)
- (5, 2)
- (6, 9)
To determine if this relation is a function, we will check whether each [tex]\(x\)[/tex] value is unique:
- [tex]\(-4\)[/tex] maps to [tex]\(0\)[/tex]
- [tex]\(-3\)[/tex] maps to [tex]\(9\)[/tex]
- [tex]\(5\)[/tex] maps to [tex]\(2\)[/tex]
- [tex]\(6\)[/tex] maps to [tex]\(9\)[/tex]
We observe that each [tex]\(x\)[/tex] value in the table is unique and maps to exactly one [tex]\(y\)[/tex] value. No [tex]\(x\)[/tex] value is repeated.
Since there are no repeated [tex]\(x\)[/tex] values, and each [tex]\(x\)[/tex] maps to only one [tex]\(y\)[/tex], this relation satisfies the definition of a function.
Thus, none of the options (A, B, C, D) presented in this problem is not a function. Therefore, the correct answer to the question "which relation is NOT a function" in this context is:
None of the given relations are not functions, as each relation in the provided table is indeed a function. The answer choices (A, B, C) do not correspond to any non-function relation.
