To determine which of the given relations represents a function, we must understand that in a function, each element from the domain (the set of all first coordinates) must be paired with exactly one element from the range (the set of all second coordinates). In other words, no two pairs in the relation should have the same first element with different second elements.
Let's analyze each relation step-by-step:
1. Relation 1: [tex]\(\{(0,0), (2,3), (2,5), (6,6)\}\)[/tex]
- First elements (domain): [tex]\(\{0, 2, 2, 6\}\)[/tex]
- Notice that the element '2' appears as the first element in both [tex]\((2,3)\)[/tex] and [tex]\((2,5)\)[/tex]. This means that the domain element '2' is associated with two different range elements (3 and 5). Therefore, this relation does not represent a function.
2. Relation 2: [tex]\(\{(3,5), (8,4), (10,11), (10,6)\}\)[/tex]
- First elements (domain): [tex]\(\{3, 8, 10, 10\}\)[/tex]
- Notice that the element '10' appears as the first element in both [tex]\((10,11)\)[/tex] and [tex]\((10,6)\)[/tex]. This means that the domain element '10' is associated with two different range elements (11 and 6). Therefore, this relation does not represent a function.
3. Relation 3: [tex]\(\{(-2,2), (0,2), (7,2), (11,2)\}\)[/tex]
- First elements (domain): [tex]\(\{-2, 0, 7, 11\}\)[/tex]
- Each first element is unique (none of the first elements are repeated). This means that every domain element is associated with exactly one range element. Therefore, this relation does represent a function.
4. Relation 4: [tex]\(\{(13,2), (13,3), (13,4), (13,5)\}\)[/tex]
- First elements (domain): [tex]\(\{13, 13, 13, 13\}\)[/tex]
- Notice that the element '13' appears as the first element in all the pairs [tex]\((13,2), (13,3), (13,4), (13,5)\)[/tex]. This means that the domain element '13' is associated with multiple range elements (2, 3, 4, 5). Therefore, this relation does not represent a function.
Based on the above analysis, we can conclude:
The relation [tex]\(\{(-2,2), (0,2), (7,2), (11,2)\}\)[/tex] is the only one that represents a function.
