To find the domain and range for the relations [tex]\( R \)[/tex] and [tex]\( Q \)[/tex], we will extract this information step-by-step.
### Relation [tex]\( R \)[/tex]
Given table for the 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]
#### Domain of [tex]\( R \)[/tex]
The domain is the set of all possible [tex]\( x \)[/tex]-values in the relation. From the table, the [tex]\( x \)[/tex]-values are:
-3, -1, 1, -1
Since the domain is a set, we eliminate duplicates. Therefore, the domain of [tex]\( R \)[/tex] is:
[tex]\[
\text{Domain of } R = \{-3, -1, 1\}
\][/tex]
#### Range of [tex]\( R \)[/tex]
The range is the set of all possible [tex]\( y \)[/tex]-values in the relation. From the table, the [tex]\( y \)[/tex]-values are:
5, 2, -1, 4
Eliminating duplicates, we get:
[tex]\[
\text{Range of } R = \{5, 2, -1, 4\}
\][/tex]
### Relation [tex]\( Q \)[/tex]
Given ordered pairs for the relation [tex]\( Q \)[/tex]:
[tex]\[
Q = \{(-2, 4), (0, 2), (-1, 3), (4, -2)\}
\][/tex]
#### Domain of [tex]\( Q \)[/tex]
The domain is the set of all possible [tex]\( x \)[/tex]-values in the relation. From the ordered pairs, the [tex]\( x \)[/tex]-values are:
-2, 0, -1, 4
Hence, the domain of [tex]\( Q \)[/tex] is:
[tex]\[
\text{Domain of } Q = \{-2, 0, -1, 4\}
\][/tex]
#### Range of [tex]\( Q \)[/tex]
The range is the set of all possible [tex]\( y \)[/tex]-values in the relation. From the ordered pairs, the [tex]\( y \)[/tex]-values are:
4, 2, 3, -2
Thus, the range of [tex]\( Q \)[/tex] is:
[tex]\[
\text{Range of } Q = \{4, 2, 3, -2\}
\][/tex]
### Final Results
[tex]\[
\text{Domain of } R = \{-3, -1, 1\}
\][/tex]
[tex]\[
\text{Range of } R = \{5, 2, -1, 4\}
\][/tex]
[tex]\[
\text{Domain of } Q = \{-2, 0, -1, 4\}
\][/tex]
[tex]\[
\text{Range of } Q = \{4, 2, 3, -2\}
\][/tex]
