To determine for which value of [tex]\( c \)[/tex] the relation is a function, we need to ensure that each x-coordinate in the set of points is unique. The relation will be a function if and only if no x-coordinate is repeated.
Given the set of points:
[tex]\[
\{(2,8), (12,3), (c, 4), (-1,8), (0,3)\}
\][/tex]
We consider the possible values for [tex]\( c \)[/tex]:
- [tex]\( c = -1 \)[/tex]
- [tex]\( c = 1 \)[/tex]
- [tex]\( c = 2 \)[/tex]
- [tex]\( c = 12 \)[/tex]
Let's analyze each case to check for uniqueness in the x-coordinates.
1. If [tex]\( c = -1 \)[/tex], the set becomes:
[tex]\[
\{(2,8), (12,3), (-1, 4), (-1, 8), (0,3)\}
\][/tex]
Here, the x-coordinate [tex]\(-1\)[/tex] is repeated. Hence, this is not a function.
2. If [tex]\( c = 1 \)[/tex], the set becomes:
[tex]\[
\{(2,8), (12,3), (1, 4), (-1, 8), (0,3)\}
\][/tex]
Here, all x-coordinates are unique: [tex]\(2, 12, 1, -1, 0\)[/tex]. Hence, this is a function.
3. If [tex]\( c = 2 \)[/tex], the set becomes:
[tex]\[
\{(2,8), (12,3), (2, 4), (-1, 8), (0,3)\}
\][/tex]
Here, the x-coordinate [tex]\(2\)[/tex] is repeated. Hence, this is not a function.
4. If [tex]\( c = 12 \)[/tex], the set becomes:
[tex]\[
\{(2,8), (12,3), (12, 4), (-1, 8), (0,3)\}
\][/tex]
Here, the x-coordinate [tex]\(12\)[/tex] is repeated. Hence, this is not a function.
From the above analysis, the only value of [tex]\( c \)[/tex] that makes the relation a function is [tex]\( c = 1 \)[/tex].
Thus, the value of [tex]\( c \)[/tex] for which the relation is a function is [tex]\(\boxed{1}\)[/tex].
