Answer :
To determine the domain and range of the relation [tex]\( R \)[/tex] given by the pairs [tex]\((x, y)\)[/tex] in the table, we follow these steps:
### Step-by-Step Solution:
1. Identify the [tex]\( x \)[/tex]-values:
From the table, extract the [tex]\( x \)[/tex]-values from each pair [tex]\((x, y)\)[/tex]:
[tex]\[ x: -3, -1, 1, -1 \][/tex]
2. Determine the domain:
The domain of a relation consists of all unique [tex]\( x \)[/tex]-values. List the unique [tex]\( x \)[/tex]-values from the previous step:
[tex]\[ \text{Domain} = \{-3, -1, 1\} \][/tex]
In set notation, this is:
[tex]\[ \text{Domain} = \{-3, -1, 1\} \][/tex]
3. Identify the [tex]\( y \)[/tex]-values:
From the table, extract the [tex]\( y \)[/tex]-values from each pair [tex]\((x, y)\)[/tex]:
[tex]\[ y: 5, 2, -1, 4 \][/tex]
4. Determine the range:
The range of a relation consists of all unique [tex]\( y \)[/tex]-values. List the unique [tex]\( y \)[/tex]-values from the previous step:
[tex]\[ \text{Range} = \{5, 2, -1, 4\} \][/tex]
In set notation, this is:
[tex]\[ \text{Range} = \{5, 2, -1, 4\} \][/tex]
### Answer:
- Domain: [tex]\(\{1, -3, -1\}\)[/tex]
- Range: [tex]\(\{2, 4, 5, -1\}\)[/tex]
Thus, we have identified the domain as [tex]\(\{1, -3, -1\}\)[/tex] and the range as [tex]\(\{2, 4, 5, -1\}\)[/tex].
### Step-by-Step Solution:
1. Identify the [tex]\( x \)[/tex]-values:
From the table, extract the [tex]\( x \)[/tex]-values from each pair [tex]\((x, y)\)[/tex]:
[tex]\[ x: -3, -1, 1, -1 \][/tex]
2. Determine the domain:
The domain of a relation consists of all unique [tex]\( x \)[/tex]-values. List the unique [tex]\( x \)[/tex]-values from the previous step:
[tex]\[ \text{Domain} = \{-3, -1, 1\} \][/tex]
In set notation, this is:
[tex]\[ \text{Domain} = \{-3, -1, 1\} \][/tex]
3. Identify the [tex]\( y \)[/tex]-values:
From the table, extract the [tex]\( y \)[/tex]-values from each pair [tex]\((x, y)\)[/tex]:
[tex]\[ y: 5, 2, -1, 4 \][/tex]
4. Determine the range:
The range of a relation consists of all unique [tex]\( y \)[/tex]-values. List the unique [tex]\( y \)[/tex]-values from the previous step:
[tex]\[ \text{Range} = \{5, 2, -1, 4\} \][/tex]
In set notation, this is:
[tex]\[ \text{Range} = \{5, 2, -1, 4\} \][/tex]
### Answer:
- Domain: [tex]\(\{1, -3, -1\}\)[/tex]
- Range: [tex]\(\{2, 4, 5, -1\}\)[/tex]
Thus, we have identified the domain as [tex]\(\{1, -3, -1\}\)[/tex] and the range as [tex]\(\{2, 4, 5, -1\}\)[/tex].
