To determine which relation defined by a set of ordered pairs is a function, we need to check that each input (first element of the pair) maps to exactly one output (second element of the pair). In other words, for the relation to be a function, each [tex]\( x \)[/tex]-value must be unique in the set.
Let's analyze each set of pairs one by one:
1. [tex]\(\{(6,2),(6,3),(5,4),(6,6),(7,6)\}\)[/tex]
- The [tex]\( x \)[/tex]-values are: [tex]\[ \{6, 6, 5, 6, 7\} \][/tex]
- Here, the [tex]\( x \)[/tex]-value 6 appears multiple times (with different [tex]\( y \)[/tex]-values). Therefore, this is not a function.
2. [tex]\(\{(2,6),(3,6),(4,5),(6,6),(6,7)\}\)[/tex]
- The [tex]\( x \)[/tex]-values are: [tex]\[ \{2, 3, 4, 6, 6\} \][/tex]
- Here, the [tex]\( x \)[/tex]-value 6 appears multiple times (with different [tex]\( y \)[/tex]-values). Thus, this is not a function.
3. [tex]\(\{(-2,6),(3,-6),(-4,5),(6,-6),(6,-7)\}\)[/tex]
- The [tex]\( x \)[/tex]-values are: [tex]\[ \{-2, 3, -4, 6, 6\} \][/tex]
- Here, the [tex]\( x \)[/tex]-value 6 appears multiple times (with different [tex]\( y \)[/tex]-values). Consequently, this is not a function.
4. [tex]\(\{(2,6),(3,6),(4,5),(6,6),(7,6)\}\)[/tex]
- The [tex]\( x \)[/tex]-values are: [tex]\[ \{2, 3, 4, 6, 7\} \][/tex]
- Here, all [tex]\( x \)[/tex]-values are unique. No [tex]\( x \)[/tex]-value is repeated. Therefore, this is a function.
Conclusion:
The set [tex]\(\{(2,6),(3,6),(4,5),(6,6),(7,6)\}\)[/tex] defines a relation that is a function, because each [tex]\( x \)[/tex]-value corresponds to exactly one [tex]\( y \)[/tex]-value.
