To determine which of the given relations is a function, we need to check if each relation satisfies the definition of a function. A relation is a function if every element of the domain (the set of all first components of ordered pairs) is mapped to exactly one element in the range (the set of all second components of ordered pairs).
Let's analyze each relation one by one:
1. [tex]\(\{(9,-4),(-2,11),(-4,6),(9,-3)\}\)[/tex]
- Domain: [tex]\(\{9, -2, -4, 9\}\)[/tex]
- The element 9 appears twice in the domain, and it is associated with two different values: -4 and -3.
- Since 9 maps to more than one value, this relation is not a function.
2. [tex]\(\{(2,1),(-3,12),(6,12),(-2,-4)\}\)[/tex]
- Domain: [tex]\(\{2, -3, 6, -2\}\)[/tex]
- All elements in the domain are unique (2 maps to 1, -3 maps to 12, 6 maps to 12, and -2 maps to -4).
- Each element of the domain maps to one and only one element in the range.
- Therefore, this relation is a function.
3. [tex]\(\{(9,11),(6,-3),(7,4),(8,4)\}\)[/tex]
- Domain: [tex]\(\{9, 6, 7, 8\}\)[/tex]
- All elements in the domain are unique (9 maps to 11, 6 maps to -3, 7 maps to 4, and 8 maps to 4).
- Each element of the domain maps to one and only one element in the range.
- Therefore, this relation is a function.
4. [tex]\(\{(5,3),(9,7),(-1,2),(6,3)\}\)[/tex]
- Domain: [tex]\(\{5, 9, -1, 6\}\)[/tex]
- All elements in the domain are unique (5 maps to 3, 9 maps to 7, -1 maps to 2, and 6 maps to 3).
- Each element of the domain maps to one and only one element in the range.
- Therefore, this relation is a function.
5. [tex]\(\{(4,9),(2,-5),(-3,-4),(2,4)\}\)[/tex]
- Domain: [tex]\(\{4, 2, -3, 2\}\)[/tex]
- The element 2 appears twice in the domain, and it is associated with two different values: -5 and 4.
- Since 2 maps to more than one value, this relation is not a function.
Based on this analysis, the relations that are functions are:
[tex]\[
\{(2,1),(-3,12),(6,12),(-2,-4)\}, \{(9,11),(6,-3),(7,4),(8,4)\}, \{(5,3),(9,7),(-1,2),(6,3)\}
\][/tex]
Therefore, the correct answer choices are:
[tex]\[
\boxed{\{(2,1),(-3,12),(6,12),(-2,-4)\}}, \boxed{\{(9,11),(6,-3),(7,4),(8,4)\}}, \boxed{\{(5,3),(9,7),(-1,2),(6,3)\}}
\][/tex]
