For the relation [tex]\(\{(2,8),(12,3),(c,4),(-1,8),(0,3)\}\)[/tex] to be a function, each input (x-value) must map to exactly one output (y-value).
To determine this, we need to ensure that each x-value is unique because a function cannot have the same x-value associated with multiple y-values.
The given relation includes the following points:
[tex]\[
(2,8), (12,3), (c,4), (-1,8), (0,3)
\][/tex]
We can list the x-values present in the points:
[tex]\[
2, 12, -1, 0
\][/tex]
We need to choose a value for [tex]\(c\)[/tex] from the provided options: [tex]\(-1, 1, 2, 12\)[/tex].
Now, we check each option:
1. If [tex]\(c = -1\)[/tex], the x-values would be [tex]\(2, 12, -1, -1, 0\)[/tex]. The x-value [tex]\(-1\)[/tex] repeats, so [tex]\(-1\)[/tex] cannot be an option for [tex]\(c\)[/tex].
2. If [tex]\(c = 1\)[/tex], the x-values would be [tex]\(2, 12, 1, -1, 0\)[/tex]. All x-values are unique, so [tex]\(1\)[/tex] is a valid option for [tex]\(c\)[/tex].
3. If [tex]\(c = 2\)[/tex], the x-values would be [tex]\(2, 12, 2, -1, 0\)[/tex]. The x-value [tex]\(2\)[/tex] repeats, so [tex]\(2\)[/tex] cannot be an option for [tex]\(c\)[/tex].
4. If [tex]\(c = 12\)[/tex], the x-values would be [tex]\(2, 12, 12, -1, 0\)[/tex]. The x-value [tex]\(12\)[/tex] repeats, so [tex]\(12\)[/tex] cannot be an option for [tex]\(c\)[/tex].
Thus, the only value for [tex]\(c\)[/tex] that ensures the relation is a function, with all x-values unique, is:
[tex]\[
c = 1
\][/tex]
