Let's break down the solution step-by-step to find the domains and ranges for the relations [tex]\( R \)[/tex] and [tex]\( Q \)[/tex].
### Relation [tex]\( R \)[/tex]:
Given the relation [tex]\( R \)[/tex] shown in the table:
| [tex]\( x \)[/tex] | [tex]\( y \)[/tex] |
|----------|---------|
| -3 | 5 |
| -1 | 2 |
| 1 | -1 |
| -1 | 4 |
Domain of [tex]\( R \)[/tex]:
The domain is the set of all unique [tex]\( x \)[/tex]-values (input values) from the relation [tex]\( R \)[/tex].
From the table, the [tex]\( x \)[/tex]-values are: [tex]\(-3, -1, 1, -1\)[/tex]. The unique [tex]\( x \)[/tex]-values form the set:
[tex]\[ \text{Domain of } R = \{-3, -1, 1\} \][/tex]
Range of [tex]\( R \)[/tex]:
The range is the set of all unique [tex]\( y \)[/tex]-values (output values) from the relation [tex]\( R \)[/tex].
From the table, the [tex]\( y \)[/tex]-values are: [tex]\(5, 2, -1, 4\)[/tex]. The unique [tex]\( y \)[/tex]-values form the set:
[tex]\[ \text{Range of } R = \{5, 2, -1, 4\} \][/tex]
### Relation [tex]\( Q \)[/tex]:
Given the relation [tex]\( Q \)[/tex] as a set of ordered pairs:
[tex]\[ Q = \{(-2, 4), (0, 2), (-1, 3), (4, -2)\} \][/tex]
Domain of [tex]\( Q \)[/tex]:
The domain is the set of all unique [tex]\( x \)[/tex]-values (input values) from the relation [tex]\( Q \)[/tex].
The [tex]\( x \)[/tex]-values from the pairs are: [tex]\(-2, 0, -1, 4\)[/tex]. The unique [tex]\( x \)[/tex]-values form the set:
[tex]\[ \text{Domain of } Q = \{-2, 0, -1, 4\} \][/tex]
Range of [tex]\( Q \)[/tex]:
The range is the set of all unique [tex]\( y \)[/tex]-values (output values) from the relation [tex]\( Q \)[/tex].
The [tex]\( y \)[/tex]-values from the pairs are: [tex]\(4, 2, 3, -2\)[/tex]. The unique [tex]\( y \)[/tex]-values form the set:
[tex]\[ \text{Range of } Q = \{4, 2, 3, -2\} \][/tex]
### Summary:
- Domain of [tex]\( R \)[/tex]: [tex]\(\{-3, -1, 1\}\)[/tex]
- Range of [tex]\( R \)[/tex]: [tex]\(\{5, 2, -1, 4\}\)[/tex]
- Domain of [tex]\( Q \)[/tex]: [tex]\(\{-2, 0, -1, 4\}\)[/tex]
- Range of [tex]\( Q \)[/tex]: [tex]\(\{4, 2, 3, -2\}\)[/tex]
So, filling in the blanks:
Domain of [tex]\( R \)[/tex]: [tex]\(\{-3, -1, 1\}\)[/tex]
Range of [tex]\( R \)[/tex]: [tex]\(\{5, 2, -1, 4\}\)[/tex]
Domain of [tex]\( Q \)[/tex]: [tex]\(\{-2, 0, -1, 4\}\)[/tex]
Range of [tex]\( Q \)[/tex]: [tex]\(\{4, 2, 3, -2\}\)[/tex]
