Let's analyze the two given relations, [tex]\( R \)[/tex] and [tex]\( Q \)[/tex], step by step to find their domains and ranges.
Relation [tex]\( R \)[/tex]:
We are provided with a table of values for relation [tex]\( R \)[/tex]:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
-3 & 5 \\
\hline
-1 & 2 \\
\hline
1 & -1 \\
\hline
-1 & 4 \\
\hline
\end{tabular}
\][/tex]
1. Domain of [tex]\( R \)[/tex]: The domain of a relation is the set of all unique [tex]\( x \)[/tex]-values.
From the table, the [tex]\( x \)[/tex]-values are: [tex]\(-3, -1, 1, -1\)[/tex].
After removing duplicates, we get the domain: [tex]\(\{-3, -1, 1\}\)[/tex].
2. Range of [tex]\( R \)[/tex]: The range of a relation is the set of all unique [tex]\( y \)[/tex]-values.
From the table, the [tex]\( y \)[/tex]-values are: [tex]\(5, 2, -1, 4\)[/tex].
There are no duplicates, so the range is: [tex]\(\{5, 2, -1, 4\}\)[/tex].
Thus, for relation [tex]\( R \)[/tex], the domain and range are:
- Domain of [tex]\( R \)[/tex]: [tex]\(\{1, -3, -1\}\)[/tex] (Note: Set notation orders elements uniquely, hence ordering may vary)
- Range of [tex]\( R \)[/tex]: [tex]\(\{2, 4, 5, -1\}\)[/tex]
Relation [tex]\( Q \)[/tex]:
Relation [tex]\( Q \)[/tex] is given as a list of ordered pairs:
[tex]\[
Q = \{(-2,4), (0,2), (-1,3), (4,-2)\}
\][/tex]
1. Domain of [tex]\( Q \)[/tex]: The domain of a relation is the set of all unique [tex]\( x \)[/tex]-values.
From the ordered pairs, the [tex]\( x \)[/tex]-values are: [tex]\(-2, 0, -1, 4\)[/tex].
So, the domain is: [tex]\(\{-2, 0, -1, 4\}\)[/tex].
2. Range of [tex]\( Q \)[/tex]: The range of a relation is the set of all unique [tex]\( y \)[/tex]-values.
From the ordered pairs, the [tex]\( y \)[/tex]-values are: [tex]\(4, 2, 3, -2\)[/tex].
So, the range is: [tex]\(\{4, 2, 3, -2\}\)[/tex].
Thus, for relation [tex]\( Q \)[/tex], the domain and range are:
- Domain of [tex]\( Q \)[/tex]: [tex]\(\{0, 4, -2, -1\}\)[/tex] (Note: Set notation orders elements uniquely, hence ordering may vary)
- Range of [tex]\( Q \)[/tex]: [tex]\(\{2, 3, 4, -2\}\)[/tex]
To summarize everything together:
- Domain of [tex]\( R \)[/tex]: [tex]\(\{1, -3, -1\}\)[/tex]
- Range of [tex]\( R \)[/tex]: [tex]\(\{2, 4, 5, -1\}\)[/tex]
- Domain of [tex]\( Q \)[/tex]: [tex]\(\{0, 4, -2, -1\}\)[/tex]
- Range of [tex]\( Q \)[/tex]: [tex]\(\{2, 3, 4, -2\}\)[/tex]
