To determine if a given table represents a function, we must ensure that each [tex]\( x \)[/tex]-value corresponds to exactly one [tex]\( y \)[/tex]-value. Here is the provided table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
2 & 1 \\
\hline
4 & 3 \\
\hline
6 & 3 \\
\hline
8 & 4 \\
\hline
10 & 6 \\
\hline
\end{array}
\][/tex]
Let's analyze the table step-by-step:
1. List the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] pairs:
- [tex]\((2, 1)\)[/tex]
- [tex]\((4, 3)\)[/tex]
- [tex]\((6, 3)\)[/tex]
- [tex]\((8, 4)\)[/tex]
- [tex]\((10, 6)\)[/tex]
2. Check the [tex]\( x \)[/tex]-values for uniqueness:
- The [tex]\( x \)[/tex]-values are: [tex]\(2, 4, 6, 8, 10\)[/tex].
- Each of these [tex]\( x \)[/tex]-values appears only once in the table.
3. Verify the [tex]\( x \)[/tex]-value mapping:
- For each [tex]\( x \)[/tex]-value, we need to see if it corresponds to exactly one [tex]\( y \)[/tex]-value.
- From the pairs listed, we see:
- [tex]\(2\)[/tex] maps to [tex]\(1\)[/tex]
- [tex]\(4\)[/tex] maps to [tex]\(3\)[/tex]
- [tex]\(6\)[/tex] maps to [tex]\(3\)[/tex]
- [tex]\(8\)[/tex] maps to [tex]\(4\)[/tex]
- [tex]\(10\)[/tex] maps to [tex]\(6\)[/tex]
4. Conclusion:
- There are no [tex]\( x \)[/tex]-values that map to more than one [tex]\( y \)[/tex]-value.
- Each [tex]\( x \)[/tex]-value has a unique corresponding [tex]\( y \)[/tex]-value.
- Despite some [tex]\( y \)[/tex]-values being the same (e.g., [tex]\(3\)[/tex] for both [tex]\(4\)[/tex] and [tex]\(6\)[/tex]), this does not violate the definition of a function, which only requires that each [tex]\( x \)[/tex]-value is paired with a single [tex]\( y \)[/tex]-value.
Therefore, the table does represent a function because every [tex]\( x \)[/tex]-value corresponds to exactly one [tex]\( y \)[/tex]-value.
The correct answer is:
C. Yes, because every [tex]\( x \)[/tex]-value corresponds to exactly one [tex]\( y \)[/tex]-value.
