Let's analyze the given relations [tex]\( R \)[/tex] and [tex]\( Q \)[/tex] to determine their domains and ranges.
### Relation [tex]\( R \)[/tex]:
The table below shows the relation [tex]\( R \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3 & 5 \\
\hline
-1 & 2 \\
\hline
1 & -1 \\
\hline
-1 & 4 \\
\hline
\end{array}
\][/tex]
Domain: The domain of a relation is the set of all unique [tex]\( x \)[/tex]-values in the relation. From the table, we have the [tex]\( x \)[/tex]-values: [tex]\(-3\)[/tex], [tex]\(-1\)[/tex], [tex]\(1\)[/tex], and [tex]\(-1\)[/tex]. Taking the unique values, we get the domain as:
[tex]\[
\text{Domain} = \{-3, -1, 1\}
\][/tex]
Let's list them distinctly:
[tex]\[
\{-3, -1, 1\}
\][/tex]
Range: The range of a relation is the set of all unique [tex]\( y \)[/tex]-values in the relation. From the table, we have the [tex]\( y \)[/tex]-values: [tex]\(5\)[/tex], [tex]\(2\)[/tex], [tex]\(-1\)[/tex], and [tex]\(4\)[/tex]. Taking the unique values, we get the range as:
[tex]\[
\text{Range} = \{5, 2, -1, 4\}
\][/tex]
Let's list them distinctly:
[tex]\[
\{2, 4, 5, -1\}
\][/tex]
### Relation [tex]\( Q \)[/tex]:
The relation [tex]\( Q \)[/tex] is given as a set of ordered pairs:
[tex]\[
Q=\{(-2,4),(0,2),(-1,3),(4,-2)\}
\][/tex]
Domain: The domain of relation [tex]\( Q \)[/tex] is the set of all unique [tex]\( x \)[/tex]-values in the ordered pairs. From the pairs, we have the [tex]\( x \)[/tex]-values: [tex]\(-2\)[/tex], [tex]\(0\)[/tex], [tex]\(-1\)[/tex], and [tex]\(4\)[/tex]. Taking the unique values, we get the domain as:
[tex]\[
\text{Domain}_Q = \{-2, 0, -1, 4\}
\][/tex]
Let's list them distinctly:
[tex]\[
\{0, 4, -2, -1\}
\][/tex]
Range: The range of relation [tex]\( Q \)[/tex] is the set of all unique [tex]\( y \)[/tex]-values in the ordered pairs. From the pairs, we have the [tex]\( y \)[/tex]-values: [tex]\(4\)[/tex], [tex]\(2\)[/tex], [tex]\(3\)[/tex], and [tex]\(-2\)[/tex]. Taking the unique values, we get the range as:
[tex]\[
\text{Range}_Q = \{4, 2, 3, -2\}
\][/tex]
Let's list them distinctly:
[tex]\[
\{2, 3, 4, -2\}
\][/tex]
### Summary:
For the relation [tex]\( R \)[/tex]:
- Domain: [tex]\([1, -3, -1]\)[/tex]
- Range: [tex]\([2, 4, 5, -1]\)[/tex]
For the relation [tex]\( Q \)[/tex]:
- Domain: [tex]\([0, 4, -2, -1]\)[/tex]
- Range: [tex]\([2, 3, 4, -2]\)[/tex]
