To determine whether the table represents a function, we need to check if each [tex]\( x \)[/tex]-value in the table corresponds to exactly one [tex]\( y \)[/tex]-value. A crucial part of defining a function is that for every input [tex]\( x \)[/tex], there must be a unique output [tex]\( y \)[/tex].
Let's examine the given table:
[tex]\[
\begin{tabular}{|l|l|}
\hline x & y \\
\hline 2 & 4 \\
\hline 3 & 5 \\
\hline 3 & 6 \\
\hline 4 & 7 \\
\hline 5 & 7 \\
\hline
\end{tabular}
\][/tex]
We'll list the pairs and inspect them to see if any [tex]\( x \)[/tex]-value maps to more than one [tex]\( y \)[/tex]-value:
1. [tex]\( (2, 4) \)[/tex] — [tex]\( x = 2 \)[/tex] maps to [tex]\( y = 4 \)[/tex].
2. [tex]\( (3, 5) \)[/tex] — [tex]\( x = 3 \)[/tex] maps to [tex]\( y = 5 \)[/tex].
3. [tex]\( (3, 6) \)[/tex] — [tex]\( x = 3 \)[/tex] maps to [tex]\( y = 6 \)[/tex].
4. [tex]\( (4, 7) \)[/tex] — [tex]\( x = 4 \)[/tex] maps to [tex]\( y = 7 \)[/tex].
5. [tex]\( (5, 7) \)[/tex] — [tex]\( x = 5 \)[/tex] maps to [tex]\( y = 7 \)[/tex].
Next, let's analyze the pairs to identify any [tex]\( x \)[/tex]-values that correspond to multiple [tex]\( y \)[/tex]-values. We observe that the [tex]\( x \)[/tex]-value [tex]\( 3 \)[/tex] maps to two different [tex]\( y \)[/tex]-values (5 and 6):
- [tex]\( 3 \rightarrow 5 \)[/tex]
- [tex]\( 3 \rightarrow 6 \)[/tex]
This relationship breaks the rule of functions where an [tex]\( x \)[/tex]-value cannot correspond to more than one [tex]\( y \)[/tex]-value. Therefore, this table does not represent a function.
### Conclusion:
The table indicates that [tex]\( x = 3 \)[/tex] corresponds to two different [tex]\( y \)[/tex]-values (5 and 6). This violates the definition of a function.
The correct choice is:
- D. No, because one [tex]\( x \)[/tex]-value corresponds to two different [tex]\( y \)[/tex]-values.
