Is the relation in the table a function? Why or why not?

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
-5 & 2 \\
\hline
-3 & -8 \\
\hline
-3 & -4 \\
\hline
2 & 7 \\
\hline
6 & 4 \\
\hline
5 & 11 \\
\hline
\end{tabular}

A. Yes. Because all the domain and range values are integers.

B. No. Because the range cannot have negative values.

C. No. Because the domain cannot have negative values.

D. No. Because the same domain value produces two different range values.



Answer :

To determine whether the relation presented in the table is a function, we need to address the key property of a function: each input (or domain value) must map to exactly one output (or range value).

Here's the given table in textual form:

[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -5 & 2 \\ \hline -3 & -8 \\ \hline -3 & -4 \\ \hline 2 & 7 \\ \hline 6 & 4 \\ \hline 5 & 11 \\ \hline \end{array} \][/tex]

### Step-by-Step Analysis:

1. Identify Domain (x-values) and Range (y-values):
- Domain: [tex]\(\{-5, -3, 2, 6, 5\}\)[/tex]
- Range: [tex]\(\{2, -8, -4, 7, 4, 11\}\)[/tex]

2. Check For Unique Mapping:
- For [tex]\(x = -5\)[/tex], [tex]\(y = 2\)[/tex]
- For [tex]\(x = -3\)[/tex], there are two [tex]\(y\)[/tex] values ([tex]\(-8\)[/tex] and [tex]\(-4\)[/tex]). This violates the definition of a function.
- For [tex]\(x = 2\)[/tex], [tex]\(y = 7\)[/tex]
- For [tex]\(x = 6\)[/tex], [tex]\(y = 4\)[/tex]
- For [tex]\(x = 5\)[/tex], [tex]\(y = 11\)[/tex]

3. Determine Whether the Relation is a Function:
- Upon closer inspection, the value [tex]\(x = -3\)[/tex] maps to two different [tex]\(y\)[/tex] values ([tex]\(-8\)[/tex] and [tex]\(-4\)[/tex]), which means that a single input leads to two different outputs.

Thus, the relation does not satisfy the definition of a function because [tex]\(x = -3\)[/tex] produces two different values in the range. Therefore, the correct conclusion is:

D. No. Because the same domain value produces two different range values.