The relation [tex]\( R \)[/tex] is shown in the table below.

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
-3 & 5 \\
\hline
-1 & 2 \\
\hline
1 & -1 \\
\hline
-1 & 4 \\
\hline
\end{tabular}

Domain: [tex]\(\{-3, -1, 1\}\)[/tex]

Range: [tex]\(\{5, 2, -1, 4\}\)[/tex]



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].