Question 22 of 25

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 4 & 3 \\
\hline 6 & 3 \\
\hline 8 & 4 \\
\hline 10 & 6 \\
\hline
\end{tabular}

A. No, because two of the [tex]$y$[/tex] values are the same.
B. Yes, because all of the [tex]$x$[/tex]-values correspond to one or more [tex]$y$[/tex]-values.
C. Yes, because every [tex]$x$[/tex]-value corresponds to exactly one [tex]$y$[/tex]-value.
D. No, because one [tex]$x$[/tex]-value corresponds to two different [tex]$y$[/tex]-values.



Answer :

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.