To determine for which value of [tex]\( c \)[/tex] the given relation is a function, we need to recall the definition of a function. In mathematics, a relation is a function if every input (or [tex]\( x \)[/tex]-value) maps to exactly one output (or [tex]\( y \)[/tex]-value).
Given the relation:
[tex]\[ \{(2,8), (12,3), (c,4), (-1,8), (0,3)\} \][/tex]
We will check to ensure that each [tex]\( x \)[/tex]-value is unique. We need to pick a value of [tex]\( c \)[/tex] such that each [tex]\( x \)[/tex]-value (input) appears only once.
Let's list the non-[tex]\( c \)[/tex] [tex]\( x \)[/tex]-values from the given pairs:
[tex]\[ x \text{-values: } 2, 12, -1, 0 \][/tex]
To ensure that our relation is a function, [tex]\( c \)[/tex] must not be equal to any of these existing [tex]\( x \)[/tex]-values. Thus, we can't use [tex]\( c = 2, 12, -1, \)[/tex] or [tex]\( 0 \)[/tex].
Examining the provided options:
- [tex]\( c = -1 \)[/tex]: This is not valid because [tex]\(-1\)[/tex] is already an [tex]\( x \)[/tex]-value.
- [tex]\( c = 1 \)[/tex]: This is valid because [tex]\( 1 \)[/tex] is not among the existing [tex]\( x \)[/tex]-values.
- [tex]\( c = 2 \)[/tex]: This is not valid because [tex]\( 2 \)[/tex] is already an [tex]\( x \)[/tex]-value.
- [tex]\( c = 12 \)[/tex]: This is not valid because [tex]\( 12 \)[/tex] is already an [tex]\( x \)[/tex]-value.
Thus, the only value for [tex]\( c \)[/tex] that ensures all [tex]\( x \)[/tex]-values are unique, making the relation a function, is [tex]\( c = 1 \)[/tex].
So, the value of [tex]\( c \)[/tex] for which the relation is a function is:
[tex]\[ \boxed{1} \][/tex]
