To find the domain and range of the given relation [tex]\( R \)[/tex], let's analyze the relation step-by-step.
### Step 1: Identify the Relation [tex]\( R \)[/tex]
The relation [tex]\( R \)[/tex] is expressed as a set of ordered pairs which can be represented as:
[tex]\[ R = \{(-3, 5), (-1, 2), (1, -1), (-1, 4)\} \][/tex]
### Step 2: Extract the Domain
The domain of a relation consists of all the distinct [tex]\( x \)[/tex]-values from the ordered pairs.
From the relation [tex]\( R \)[/tex]:
- The [tex]\( x \)[/tex]-value in the first pair is [tex]\(-3\)[/tex].
- The [tex]\( x \)[/tex]-value in the second pair is [tex]\(-1\)[/tex].
- The [tex]\( x \)[/tex]-value in the third pair is [tex]\( 1 \)[/tex].
- The [tex]\( x \)[/tex]-value in the fourth pair is [tex]\(-1\)[/tex] (note: this is a repetition of [tex]\(-1\)[/tex] which we have already counted).
So, the distinct [tex]\( x \)[/tex]-values are:
[tex]\[ \{-3, -1, 1\} \][/tex]
The domain is therefore:
[tex]\[ \text{Domain} = \{-3, -1, 1\} \][/tex]
### Step 3: Extract the Range
The range of a relation consists of all the distinct [tex]\( y \)[/tex]-values from the ordered pairs.
From the relation [tex]\( R \)[/tex]:
- The [tex]\( y \)[/tex]-value in the first pair is [tex]\( 5 \)[/tex].
- The [tex]\( y \)[/tex]-value in the second pair is [tex]\( 2 \)[/tex].
- The [tex]\( y \)[/tex]-value in the third pair is [tex]\(-1\)[/tex].
- The [tex]\( y \)[/tex]-value in the fourth pair is [tex]\( 4 \)[/tex].
So, the distinct [tex]\( y \)[/tex]-values are:
[tex]\[ \{5, 2, -1, 4\} \][/tex]
The range is therefore:
[tex]\[ \text{Range} = \{5, 2, -1, 4\} \][/tex]
### Final Answer
- Domain: [tex]\(\{-3, -1, 1\}\)[/tex]
- Range: [tex]\(\{5, 2, -1, 4\}\)[/tex]
