Select the correct answer.

Which of the following represents a function?

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]

B.

C.

D. [tex]$\{[-7,-9),(-4,-9),(5,15),(7,19)\}$[/tex]



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.