Does this table represent a function? Why or why not?

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
2 & 1 \\
\hline
2 & 4 \\
\hline
3 & 4 \\
\hline
4 & 2 \\
\hline
5 & 5 \\
\hline
\end{tabular}

A. Yes, because there are two [tex]$x$[/tex]-values that are the same.

B. Yes, because every [tex]$x$[/tex]-value corresponds to exactly one [tex]$y$[/tex]-value.

C. No, because two of the [tex]$y$[/tex]-values are the same.

D. No, because one [tex]$x$[/tex]-value corresponds to two different [tex]$y$[/tex]-values.



Answer :

To determine if the given table represents a function, we need to recall the definition of a function. A function is defined as a relation where every input \( x \) corresponds to exactly one output \( y \). In other words, each \( x \) value must map to one and only one \( y \) value.

Let's analyze the table:

[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 2 & 1 \\ \hline 2 & 4 \\ \hline 3 & 4 \\ \hline 4 & 2 \\ \hline 5 & 5 \\ \hline \end{array} \][/tex]

1. For \( x = 2 \), we have two outputs: \( y = 1 \) and \( y = 4 \). This means that the input \( x = 2 \) corresponds to two different \( y \) values, which violates the definition of a function.
2. Other \( x \) values (3, 4, and 5) each map to a single \( y \) value.

Given this analysis, the table does not represent a function because the input \( x = 2 \) yields two different outputs. Therefore, the correct reasoning is:

D. No, because one [tex]\( x \)[/tex]-value corresponds to two different [tex]\( y \)[/tex]-values.