To determine which point could be removed to make the relation a function, we need to understand what constitutes a function. A function is a relation where each input (or [tex]\(x\)[/tex]-coordinate) has exactly one output (or [tex]\(y\)[/tex]-coordinate). In other words, no [tex]\(x\)[/tex]-coordinate should be repeated with different [tex]\(y\)[/tex]-coordinates.
Given the set of points:
[tex]\[
\{(0,2), (3,8), (-4,-2), (3,-6), (-1,8), (8,3)\}
\][/tex]
Let's examine the [tex]\(x\)[/tex]-coordinates:
[tex]\[
0, 3, -4, 3, -1, 8
\][/tex]
We notice that the [tex]\(x\)[/tex]-coordinate [tex]\(3\)[/tex] is repeated. The points with [tex]\(x=3\)[/tex] are [tex]\((3,8)\)[/tex] and [tex]\((3,-6)\)[/tex]. For this relation to be a function, we need to remove one of these points to ensure that each [tex]\(x\)[/tex]-coordinate is unique.
Thus, the candidates for removal are [tex]\((3,8)\)[/tex] or [tex]\((3,-6)\)[/tex].
We need to choose which one to remove based on the options given:
[tex]\[
(8,3), (3,-6), (-1,8), (-4,-2)
\][/tex]
Only one of the options provided matches our candidate points:
[tex]\((3,-6)\)[/tex]
Therefore, removing the point [tex]\((3,-6)\)[/tex] will make the relation a function. The detailed step-by-step reasoning leads us to conclude that:
The point [tex]\((3, -6)\)[/tex] could be removed in order to make the relation a function.
