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.
