Answer :
Let's break down each function and their respective properties step-by-step to match the given characteristics:
1. Function [tex]\( P = \{(x, y) \mid y=3\} \)[/tex]
- Domain: The domain of [tex]\(P\)[/tex] consists of all possible values of [tex]\(x\)[/tex]. Since [tex]\(y\)[/tex] is always 3 and there is no restriction on [tex]\(x\)[/tex], the domain includes all real numbers.
- Range: The range refers to all possible [tex]\(y\)[/tex]-values. For [tex]\(P\)[/tex], since [tex]\(y\)[/tex] always equals 3 regardless of [tex]\(x\)[/tex], the range is the single value {3}.
Thus, for [tex]\(P\)[/tex]:
- Domain: all real numbers
- Range: \{3\}
Therefore, the correct match is:
```
- the range and domain of [tex]\(P = \{(x, y) \mid y=3\}\)[/tex]: domain = all real numbers; range = {3}
```
2. Function [tex]\( C = \{(2,5),(2,6),(2,7)\} \)[/tex]
- Domain: The domain consists of the set of all distinct [tex]\(x\)[/tex]-values. Since [tex]\(x\)[/tex] is 2 in every ordered pair, the domain is \{2\}.
- Range: Although the range isn't explicitly asked for, it would be the distinct [tex]\( y \)[/tex]-values, which are \{5, 6, 7\} in this case.
Thus, for [tex]\(C\)[/tex]:
- Domain: \{2\}
Therefore, the correct match is:
```
- the domain set of [tex]\(C = \{(2,5),(2,6),(2,7)\}\)[/tex]: domain = {2}
```
3. Function [tex]\( E = \{(3,3),(4,4),(5,5),(6,6)\} \)[/tex]
- Range: The range consists of all distinct [tex]\(y\)[/tex]-values in these ordered pairs. Since [tex]\(y\)[/tex] takes the values 3, 4, 5, and 6, the range is \{3, 4, 5, 6\}.
Thus, for [tex]\(E\)[/tex]:
- Range: \{3, 4, 5, 6\}
Therefore, the correct match is:
```
- the range set of [tex]\(E = \{(3,3),(4,4),(5,5),(6,6)\}\)[/tex]: {3, 4, 5, 6}
```
4. Function [tex]\(F = \{(x, y) \mid x + y = 10\} \)[/tex]
- Domain: The domain consists of all possible [tex]\(x\)[/tex]-values that satisfy the equation [tex]\( x + y = 10\)[/tex] for some [tex]\( y \)[/tex]. Here, [tex]\( x \)[/tex] can be any real number.
- Range: The range consists of all possible [tex]\(y\)[/tex]-values that satisfy [tex]\( x + y = 10\)[/tex] for some [tex]\( x \)[/tex]. Here, [tex]\( y \)[/tex] can also be any real number.
Thus, for [tex]\(F\)[/tex]:
- Domain: all real numbers
- Range: all real numbers
Therefore, the correct match is:
```
- the range and domain of [tex]\(F = \{(x, y) \mid x+y=10\}\)[/tex]: domain = all real numbers; range = all real numbers
```
So, the matches are:
1. the range and domain of [tex]\( P = \{(x, y) \mid y=3\} \)[/tex]: domain = all real numbers; range = {3}
2. the domain set of [tex]\( C = \{(2,5),(2,6),(2,7)\} \)[/tex]: domain = {2}
3. the range set of [tex]\( E = \{(3,3),(4,4),(5,5),(6,6)\} \)[/tex]: {3, 4, 5, 6}
4. the range and domain of [tex]\( F = \{(x, y) \mid x+y=10\}\)[/tex]: domain = all real numbers; range = all real numbers
1. Function [tex]\( P = \{(x, y) \mid y=3\} \)[/tex]
- Domain: The domain of [tex]\(P\)[/tex] consists of all possible values of [tex]\(x\)[/tex]. Since [tex]\(y\)[/tex] is always 3 and there is no restriction on [tex]\(x\)[/tex], the domain includes all real numbers.
- Range: The range refers to all possible [tex]\(y\)[/tex]-values. For [tex]\(P\)[/tex], since [tex]\(y\)[/tex] always equals 3 regardless of [tex]\(x\)[/tex], the range is the single value {3}.
Thus, for [tex]\(P\)[/tex]:
- Domain: all real numbers
- Range: \{3\}
Therefore, the correct match is:
```
- the range and domain of [tex]\(P = \{(x, y) \mid y=3\}\)[/tex]: domain = all real numbers; range = {3}
```
2. Function [tex]\( C = \{(2,5),(2,6),(2,7)\} \)[/tex]
- Domain: The domain consists of the set of all distinct [tex]\(x\)[/tex]-values. Since [tex]\(x\)[/tex] is 2 in every ordered pair, the domain is \{2\}.
- Range: Although the range isn't explicitly asked for, it would be the distinct [tex]\( y \)[/tex]-values, which are \{5, 6, 7\} in this case.
Thus, for [tex]\(C\)[/tex]:
- Domain: \{2\}
Therefore, the correct match is:
```
- the domain set of [tex]\(C = \{(2,5),(2,6),(2,7)\}\)[/tex]: domain = {2}
```
3. Function [tex]\( E = \{(3,3),(4,4),(5,5),(6,6)\} \)[/tex]
- Range: The range consists of all distinct [tex]\(y\)[/tex]-values in these ordered pairs. Since [tex]\(y\)[/tex] takes the values 3, 4, 5, and 6, the range is \{3, 4, 5, 6\}.
Thus, for [tex]\(E\)[/tex]:
- Range: \{3, 4, 5, 6\}
Therefore, the correct match is:
```
- the range set of [tex]\(E = \{(3,3),(4,4),(5,5),(6,6)\}\)[/tex]: {3, 4, 5, 6}
```
4. Function [tex]\(F = \{(x, y) \mid x + y = 10\} \)[/tex]
- Domain: The domain consists of all possible [tex]\(x\)[/tex]-values that satisfy the equation [tex]\( x + y = 10\)[/tex] for some [tex]\( y \)[/tex]. Here, [tex]\( x \)[/tex] can be any real number.
- Range: The range consists of all possible [tex]\(y\)[/tex]-values that satisfy [tex]\( x + y = 10\)[/tex] for some [tex]\( x \)[/tex]. Here, [tex]\( y \)[/tex] can also be any real number.
Thus, for [tex]\(F\)[/tex]:
- Domain: all real numbers
- Range: all real numbers
Therefore, the correct match is:
```
- the range and domain of [tex]\(F = \{(x, y) \mid x+y=10\}\)[/tex]: domain = all real numbers; range = all real numbers
```
So, the matches are:
1. the range and domain of [tex]\( P = \{(x, y) \mid y=3\} \)[/tex]: domain = all real numbers; range = {3}
2. the domain set of [tex]\( C = \{(2,5),(2,6),(2,7)\} \)[/tex]: domain = {2}
3. the range set of [tex]\( E = \{(3,3),(4,4),(5,5),(6,6)\} \)[/tex]: {3, 4, 5, 6}
4. the range and domain of [tex]\( F = \{(x, y) \mid x+y=10\}\)[/tex]: domain = all real numbers; range = all real numbers
