Answer :
Let's carefully analyze each option to determine which one represents a function.
Definition of a function:
A function is a relation in which each element of the domain (all possible [tex]\(x\)[/tex]-values) is paired with exactly one element of the range (all possible [tex]\(y\)[/tex]-values). This means that no [tex]\(x\)[/tex]-value should be repeated with a different [tex]\(y\)[/tex]-value.
### Option A
[tex]\[ \begin{tabular}{|c|c|c|c|c|c|} \hline $x$ & 5 & -5 & 10 & 5 & -10 \\ \hline $y$ & 13 & -7 & 23 & 17 & -17 \\ \hline \end{tabular} \][/tex]
In this table, the [tex]\(x\)[/tex]-value 5 is associated with both 13 and 17. Since one [tex]\(x\)[/tex]-value maps to two different [tex]\(y\)[/tex]-values, this option does not represent a function.
### Option B
[tex]\[ \{(0, 2), (2, 0), (4, 3)\} \][/tex]
Here, each [tex]\(x\)[/tex]-value (0, 2, and 4) is paired with exactly one unique [tex]\(y\)[/tex]-value. There are no repeated [tex]\(x\)[/tex]-values with different [tex]\(y\)[/tex]-values. Therefore, this option represents a function.
### Option C
There is no Option C provided in the text.
### Option D
[tex]\[ \{(-7, -9), (-4, -9), (5, 15), (7, 19)\} \][/tex]
In this set, each [tex]\(x\)[/tex]-value (-7, -4, 5, and 7) is paired with exactly one unique [tex]\(y\)[/tex]-value. No [tex]\(x\)[/tex]-value is repeated here, which means this option also represents a function.
### Summary
Based on the given options, we find that:
- Option A is not a function because the [tex]\(x\)[/tex]-value 5 is associated with more than one [tex]\(y\)[/tex]-value.
- Option B is a function as each [tex]\(x\)[/tex]-value has a unique [tex]\(y\)[/tex]-value.
- Option D is also a function for the same reason.
Since we need to select one correct answer, we conclude that:
The correct answer is Option B.
Definition of a function:
A function is a relation in which each element of the domain (all possible [tex]\(x\)[/tex]-values) is paired with exactly one element of the range (all possible [tex]\(y\)[/tex]-values). This means that no [tex]\(x\)[/tex]-value should be repeated with a different [tex]\(y\)[/tex]-value.
### Option A
[tex]\[ \begin{tabular}{|c|c|c|c|c|c|} \hline $x$ & 5 & -5 & 10 & 5 & -10 \\ \hline $y$ & 13 & -7 & 23 & 17 & -17 \\ \hline \end{tabular} \][/tex]
In this table, the [tex]\(x\)[/tex]-value 5 is associated with both 13 and 17. Since one [tex]\(x\)[/tex]-value maps to two different [tex]\(y\)[/tex]-values, this option does not represent a function.
### Option B
[tex]\[ \{(0, 2), (2, 0), (4, 3)\} \][/tex]
Here, each [tex]\(x\)[/tex]-value (0, 2, and 4) is paired with exactly one unique [tex]\(y\)[/tex]-value. There are no repeated [tex]\(x\)[/tex]-values with different [tex]\(y\)[/tex]-values. Therefore, this option represents a function.
### Option C
There is no Option C provided in the text.
### Option D
[tex]\[ \{(-7, -9), (-4, -9), (5, 15), (7, 19)\} \][/tex]
In this set, each [tex]\(x\)[/tex]-value (-7, -4, 5, and 7) is paired with exactly one unique [tex]\(y\)[/tex]-value. No [tex]\(x\)[/tex]-value is repeated here, which means this option also represents a function.
### Summary
Based on the given options, we find that:
- Option A is not a function because the [tex]\(x\)[/tex]-value 5 is associated with more than one [tex]\(y\)[/tex]-value.
- Option B is a function as each [tex]\(x\)[/tex]-value has a unique [tex]\(y\)[/tex]-value.
- Option D is also a function for the same reason.
Since we need to select one correct answer, we conclude that:
The correct answer is Option B.
