To determine which relations represent a function, we need to recall the definition of a function. A relation is a function if every element in the domain (the set of all first components of the ordered pairs) is paired with exactly one element in the range (the set of all second components of the ordered pairs). In simpler terms, no [tex]\( x \)[/tex]-value should repeat with different [tex]\( y \)[/tex]-values.
Let's analyze each relation step-by-step:
1. Relation: [tex]\(\{(0, 0), (2, 3), (2, 5), (6, 6)\}\)[/tex]
- The domain for this relation is [tex]\(\{0, 2, 2, 6\}\)[/tex].
- The value [tex]\(2\)[/tex] appears twice, and it is associated with different values in the range ([tex]\(3\)[/tex] and [tex]\(5\)[/tex]).
Since there is a repeated [tex]\(x\)[/tex]-value that maps to different [tex]\(y\)[/tex]-values, this relation is not a function.
2. Relation: [tex]\(\{(3, 5), (8, 4), (10, 11), (10, 6)\}\)[/tex]
- The domain for this relation is [tex]\(\{3, 8, 10, 10\}\)[/tex].
- The value [tex]\(10\)[/tex] appears twice, and it is associated with different values in the range ([tex]\(11\)[/tex] and [tex]\(6\)[/tex]).
Since there is a repeated [tex]\(x\)[/tex]-value that maps to different [tex]\(y\)[/tex]-values, this relation is not a function.
3. Relation: [tex]\(\{(-2, 2), (0, 2), (7, 2), (11, 2)\}\)[/tex]
- The domain for this relation is [tex]\(\{-2, 0, 7, 11\}\)[/tex].
- All [tex]\(x\)[/tex]-values are unique, and each [tex]\(x\)[/tex]-value maps to exactly one [tex]\(y\)[/tex]-value.
Since there are no repeated [tex]\(x\)[/tex]-values, this relation is a function.
4. Relation: [tex]\(\{(13, 2), (13, 3), (13, 4), (13, 5)\}\)[/tex]
- The domain for this relation is [tex]\(\{13, 13, 13, 13\}\)[/tex].
- The value [tex]\(13\)[/tex] appears multiple times and is associated with different [tex]\(y\)[/tex]-values ([tex]\(2, 3, 4, 5\)[/tex]).
Since there is a repeated [tex]\(x\)[/tex]-value that maps to different [tex]\(y\)[/tex]-values, this relation is not a function.
Therefore, the relation [tex]\(\{(-2, 2), (0, 2), (7, 2), (11, 2)\}\)[/tex] is the only one that represents a function.
