To determine which of the given relations represents a function, we have to recall the definition of a function. A relation is a function if every input (or [tex]\( x \)[/tex]-value) maps to exactly one output (or [tex]\( y \)[/tex]-value). In simpler terms, if any [tex]\( x \)[/tex]-value is repeated in the relation, the corresponding [tex]\( y \)[/tex]-values must be the same for it to be a function.
Let's analyze each relation separately:
### Relation 1: [tex]\(\{(0,0),(2,3),(2,5),(6,6)\}\)[/tex]
- Here, the pair [tex]\((2,3)\)[/tex] and [tex]\((2,5)\)[/tex] have the same [tex]\( x \)[/tex]-value (2) with different [tex]\( y \)[/tex]-values (3 and 5). This means that the same input (2) gives two different outputs (3 and 5).
- Therefore, this relation is not a function.
### Relation 2: [tex]\(\{(3,5),(8,4),(10,11),(10,6)\}\)[/tex]
- Here, the pair [tex]\((10,11)\)[/tex] and [tex]\((10,6)\)[/tex] have the same [tex]\( x \)[/tex]-value (10) with different [tex]\( y \)[/tex]-values (11 and 6). This means the same input (10) gives two different outputs (11 and 6).
- Therefore, this relation is not a function.
### Relation 3: [tex]\(\{(-2,2),(0,2),(7,2),(11,2)\}\)[/tex]
- In this relation, each pair has a unique [tex]\( x \)[/tex]-value: [tex]\(-2\)[/tex], [tex]\(0\)[/tex], [tex]\(7\)[/tex], and [tex]\(11\)[/tex]. Each input maps to only one output.
- Therefore, this relation is a function.
### Relation 4: [tex]\(\{(13,2),(13,3),(13,4),(13,5)\}\)[/tex]
- Here, all pairs have the same [tex]\( x \)[/tex]-value (13) but different [tex]\( y \)[/tex]-values (2, 3, 4, and 5). This means that the same input (13) gives multiple outputs.
- Therefore, this relation is not a function.
After evaluating each relation, we find that the relation [tex]\(\{(-2,2),(0,2),(7,2),(11,2)\}\)[/tex] is the only one that meets the criteria of a function. Hence, the third relation represents a function.
Therefore, the answer is:
[tex]\[ \{(-2,2),(0,2),(7,2),(11,2)\} \][/tex]
