Sure, let's go step by step to determine the domain and range of the relation [tex]\( R \)[/tex].
A relation [tex]\( R \)[/tex] is given by the set of ordered pairs:
[tex]\[ R = \{(-3, -2), (-3, 0), (-1, 2), (1, 2)\} \][/tex]
Step-by-Step Solution:
### Identifying the Domain:
The domain of a relation consists of all the first elements (x-coordinates) of the ordered pairs. Let's list them out:
1. From the pair [tex]\((-3, -2)\)[/tex], the first element is [tex]\(-3\)[/tex].
2. From the pair [tex]\((-3, 0)\)[/tex], the first element is [tex]\(-3\)[/tex].
3. From the pair [tex]\((-1, 2)\)[/tex], the first element is [tex]\(-1\)[/tex].
4. From the pair [tex]\((1, 2)\)[/tex], the first element is [tex]\(1\)[/tex].
Now, let's list the unique values among these:
[tex]\[ \{-3, -3, -1, 1\} \][/tex]
Remove the duplicate [tex]\(-3\)[/tex]:
[tex]\[ \{-3, -1, 1\} \][/tex]
So, the domain is [tex]\(\{-3, -1, 1\}\)[/tex].
### Identifying the Range:
The range of a relation consists of all the second elements (y-coordinates) of the ordered pairs. Let's list them out:
1. From the pair [tex]\((-3, -2)\)[/tex], the second element is [tex]\(-2\)[/tex].
2. From the pair [tex]\((-3, 0)\)[/tex], the second element is [tex]\(0\)[/tex].
3. From the pair [tex]\((-1, 2)\)[/tex], the second element is [tex]\(2\)[/tex].
4. From the pair [tex]\((1, 2)\)[/tex], the second element is [tex]\(2\)[/tex].
Now, let's list the unique values among these:
[tex]\[ \{-2, 0, 2, 2\} \][/tex]
Remove the duplicate [tex]\(2\)[/tex]:
[tex]\[ \{-2, 0, 2\} \][/tex]
So, the range is [tex]\(\{-2, 0, 2\}\)[/tex].
### Answering the Specific Questions:
#### Domain Selections:
- [tex]\(\checkmark \quad -3\)[/tex] (yes, it is in the domain)
- [tex]\(x \quad -2\)[/tex] (no, it is not in the domain)
- [tex]\(\checkmark \quad -1\)[/tex] (yes, it is in the domain)
- [tex]\(\times \quad 0\)[/tex] (no, it is not in the domain)
- [tex]\(\checkmark \quad 1\)[/tex] (yes, it is in the domain)
- [tex]\(x \quad 2\)[/tex] (no, it is not in the domain)
#### Range Selections:
- [tex]\(\times \quad -3\)[/tex] (no, it is not in the range)
- [tex]\(\checkmark \quad -2\)[/tex] (yes, it is in the range)
