To determine the domain and range of the given sets of relations, we need to analyze the provided [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values from the table.
First, let's identify the domain and range of the relation described by the table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
-4 & -2 \\
\hline
-2 & 1 \\
\hline
1 & 4 \\
\hline
4 & 4 \\
\hline
\end{tabular}
\][/tex]
- The domain consists of all the [tex]\(x\)[/tex] values:
[tex]\[
\{-4, -2, 1, 4\}
\][/tex]
- The range consists of all the [tex]\(y\)[/tex] values:
[tex]\[
\{-2, 1, 4\}
\][/tex]
Next, let's compare the domain and range we've determined with the provided sets to match each one:
1. Domain: [tex]\(\{-2, -1, 0, 2\}\)[/tex]
Range: [tex]\(\{-4, -2, 2\}\)[/tex]
This does not match the domain [tex]\(\{-4, -2, 1, 4\}\)[/tex] or the range [tex]\(\{-2, 1, 4\}\)[/tex] of our relation.
So, the match for this set is:
[tex]\((\text{False}, \text{False})\)[/tex]
2. Domain: [tex]\(\{-4, -2, 1, 4\}\)[/tex]
Range: [tex]\(\{-2, 1, 4\}\)[/tex]
This precisely matches the domain [tex]\(\{-4, -2, 1, 4\}\)[/tex] and the range [tex]\(\{-2, 1, 4\}\)[/tex] of our relation.
So, the match for this set is:
[tex]\((\text{True}, \text{True})\)[/tex]
3. Domain: [tex]\(\{-4, -2, -1\}\)[/tex]
Range: [tex]\(\{-2, 0, 1\}\)[/tex]
This does not match the domain [tex]\(\{-4, -2, 1, 4\}\)[/tex] or the range [tex]\(\{-2, 1, 4\}\)[/tex] of our relation.
So, the match for this set is:
[tex]\((\text{False}, \text{False})\)[/tex]
Hence:
1. The pair [tex]\(\{-2, -1, 0, 2\}\)[/tex] and [tex]\(\{-4, -2, 2\}\)[/tex] does not match our relation.
2. The pair [tex]\(\{-4, -2, 1, 4\}\)[/tex] and [tex]\(\{-2, 1, 4\}\)[/tex] correctly matches our relation.
3. The pair [tex]\(\{-4, -2, -1\}\)[/tex] and [tex]\(\{-2, 0, 1\}\)[/tex] does not match our relation.
Therefore, the correct match is:
[tex]\[
\{ \{-4, -2, 1, 4\}, \{-2, 1, 4\} \}
\][/tex]
