To find the domain and range of the relation [tex]\(Q\)[/tex], we need to identify the unique first elements (domain) and unique second elements (range) of each ordered pair in the set.
The given relation is:
[tex]\[ Q = \{(-2, 4), (0, 2), (-1, 3), (4, -2)\} \][/tex]
Step 1: Determine the Domain
The domain of a relation consists of all the first elements from each ordered pair. Let’s list them:
- The first element of [tex]\((-2, 4)\)[/tex] is [tex]\(-2\)[/tex].
- The first element of [tex]\((0, 2)\)[/tex] is [tex]\(0\)[/tex].
- The first element of [tex]\((-1, 3)\)[/tex] is [tex]\(-1\)[/tex].
- The first element of [tex]\((4, -2)\)[/tex] is [tex]\(4\)[/tex].
Therefore, the domain is:
[tex]\[ \{-2, 0, -1, 4\} \][/tex]
Step 2: Determine the Range
The range of a relation consists of all the second elements from each ordered pair. Let’s list them:
- The second element of [tex]\((-2, 4)\)[/tex] is [tex]\(4\)[/tex].
- The second element of [tex]\((0, 2)\)[/tex] is [tex]\(2\)[/tex].
- The second element of [tex]\((-1, 3)\)[/tex] is [tex]\(3\)[/tex].
- The second element of [tex]\((4, -2)\)[/tex] is [tex]\(-2\)[/tex].
Therefore, the range is:
[tex]\[ \{4, 2, 3, -2\} \][/tex]
So, the domain and range of the relation [tex]\(Q\)[/tex] are:
- Domain: [tex]\( \{0, 4, -1, -2\} \)[/tex]
- Range: [tex]\( \{2, 3, 4, -2\} \)[/tex]
Therefore, the final answers are:
Domain: [tex]\(\{0, 4, -1, -2\}\)[/tex]
Range: [tex]\(\{2, 3, 4, -2\}\)[/tex]
