To determine in which sets the given relations represent a function, we need to verify if each relation maintains the definition of a function: that for every element [tex]\( x \)[/tex] in the domain, there should be exactly one corresponding element [tex]\( y \)[/tex] in the co-domain.
Let's analyze the relations one by one:
1. [tex]\(\{(4, 9), (0, -2), (0, 2), (5, 4)\}\)[/tex]
- Here, [tex]\( 0 \)[/tex] is associated with both [tex]\(-2\)[/tex] and [tex]\(2\)[/tex]. Therefore, this relation is not a function.
2. [tex]\(\{(5, -5), (5, -4), (7, -2), (3, 8)\}\)[/tex]
- Here, [tex]\( 5 \)[/tex] is associated with both [tex]\(-5\)[/tex] and [tex]\(-4\)[/tex]. Therefore, this relation is not a function.
3. [tex]\(\{(4, 3), (8, 0), (5, 2), (-5, 0)\}\)[/tex]
- Here, each [tex]\( x \)[/tex] value (4, 8, 5, -5) is unique and appears only once in the domain. Therefore, this relation is a function.
4. [tex]\(\{(6, 9), (9, -4), (6, 1), (-5, 11)\}\)[/tex]
- Here, [tex]\( 6 \)[/tex] is associated with both [tex]\(9\)[/tex] and [tex]\(1\)[/tex]. Therefore, this relation is not a function.
5. [tex]\(\{(4, 12), (2, 6), (-5, 6), (3, -2)\}\)[/tex]
- Each [tex]\( x \)[/tex] value (4, 2, -5, 3) is unique and appears only once in the domain. Therefore, this relation is a function.
Based on this analysis, the relations that are functions are:
[tex]\[
\{(4, 3), (8, 0), (5, 2), (-5, 0)\}
\{(4, 12), (2, 6), (-5, 6), (3, -2)\}
\][/tex]
Therefore, the correct answer is:
[tex]\([3, 5]\)[/tex]
