To determine whether a given relation represents a function, we need to check if each input (or x-value) maps to exactly one output (or y-value). In other words, a relation is a function if no x-value is repeated with different y-values.
Let's examine each relation step by step:
1. [tex]\(\{ (0, 0), (2, 3), (2, 5), (6, 6) \}\)[/tex]
- The x-values are: 0, 2, 2, 6.
- We observe that the x-value 2 appears twice with different y-values (3 and 5).
- Therefore, this relation does not represent a function.
2. [tex]\(\{ (3, 5), (8, 4), (10, 11), (10, 6) \}\)[/tex]
- The x-values are: 3, 8, 10, 10.
- We observe that the x-value 10 appears twice with different y-values (11 and 6).
- Therefore, this relation does not represent a function.
3. [tex]\(\{ (-2, 2), (0, 2), (7, 2), (11, 2) \}\)[/tex]
- The x-values are: -2, 0, 7, 11.
- Each x-value is unique and appears only once.
- Therefore, this relation represents a function.
4. [tex]\(\{ (13, 2), (13, 3), (13, 4), (13, 5) \}\)[/tex]
- The x-values are: 13, 13, 13, 13.
- The x-value 13 appears multiple times with different y-values (2, 3, 4, and 5).
- Therefore, this relation does not represent a function.
By analyzing each relation, we conclude that the relation [tex]\(\{ (-2, 2), (0, 2), (7, 2), (11, 2) \}\)[/tex] is the one that represents a function.
