To determine whether the given table represents a function, we need to examine the relationship between the [tex]\(x\)[/tex]-values (inputs) and [tex]\(y\)[/tex]-values (outputs). Specifically, a set of ordered pairs [tex]\((x, y)\)[/tex] represents a function if and only if each [tex]\(x\)[/tex]-value is associated with exactly one [tex]\(y\)[/tex]-value.
Let's analyze the table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
2 & 1 \\
\hline
3 & 4 \\
\hline
4 & 4 \\
\hline
5 & 2 \\
\hline
5 & 5 \\
\hline
\end{tabular}
\][/tex]
We need to check if there is any [tex]\(x\)[/tex]-value that corresponds to more than one [tex]\(y\)[/tex]-value:
- For [tex]\(x = 2\)[/tex], we have [tex]\(y = 1\)[/tex].
- For [tex]\(x = 3\)[/tex], we have [tex]\(y = 4\)[/tex].
- For [tex]\(x = 4\)[/tex], we have [tex]\(y = 4\)[/tex].
- For [tex]\(x = 5\)[/tex], we encounter a problem: [tex]\(x = 5\)[/tex] corresponds to both [tex]\(y = 2\)[/tex] and [tex]\(y = 5\)[/tex].
Since the [tex]\(x\)[/tex]-value [tex]\(5\)[/tex] corresponds to more than one [tex]\(y\)[/tex]-value ([tex]\(2\)[/tex] and [tex]\(5\)[/tex]), the table does not represent a function. This means that the criterion for a function (each [tex]\(x\)[/tex]-value corresponds to exactly one [tex]\(y\)[/tex]-value) is not met.
Therefore, the correct answer is:
C. No, because one [tex]\(x\)[/tex]-value corresponds to two different [tex]\(y\)[/tex]-values.
