To determine which of the given relations are functions, we need to ensure that each input (the first element of the ordered pair) maps to exactly one output (the second element of the ordered pair). This means no input should be repeated with a different output.
Let's analyze each relation step by step:
1. Relation: [tex]\(\{(2,-5),(-2,0),(-3,6),(2,-4)\}\)[/tex]\
- Here, the input 2 appears twice: [tex]\((2, -5)\)[/tex] and [tex]\((2, -4)\)[/tex].
- Since the input 2 maps to two different outputs (-5 and -4), this relation is not a function.
2. Relation: [tex]\(\{(-1,5),(-4,8),(-4,14),(2,6)\}\)[/tex]
- Here, the input -4 appears twice: [tex]\((-4, 8)\)[/tex] and [tex]\((-4, 14)\)[/tex].
- Since the input -4 maps to two different outputs (8 and 14), this relation is not a function.
3. Relation: [tex]\(\{(1,3),(-2,-1),(4,3),(8,1)\}\)[/tex]
- All inputs are unique: 1, -2, 4, and 8.
- Since no input is repeated, each input maps to exactly one output.
- Therefore, this relation is a function.
4. Relation: [tex]\(\{(-2,-5),(7,1),(7,-3),(4,-1)\}\)[/tex]\
- Here, the input 7 appears twice: [tex]\((7, 1)\)[/tex] and [tex]\((7, -3)\)[/tex].
- Since the input 7 maps to two different outputs (1 and -3), this relation is not a function.
5. Relation: [tex]\(\{(8,8),(4,1),(1,6),(-5,6)\}\)[/tex]\
- All inputs are unique: 8, 4, 1, and -5.
- Since no input is repeated, each input maps to exactly one output.
- Therefore, this relation is a function.
Hence, the correct choices where the relation is a function are:
[tex]\[\{(1,3),(-2,-1),(4,3),(8,1)\}\][/tex]
[tex]\[\{(8,8),(4,1),(1,6),(-5,6)\}\][/tex]
The correct answer is:
[tex]\[ \boxed{3, 5} \][/tex]
