To determine which relation represents a one-to-one function, we need to verify that each [tex]\( f(x) \)[/tex] value is unique for each [tex]\( x \)[/tex] value. In other words, no two different [tex]\( x \)[/tex] values should map to the same [tex]\( f(x) \)[/tex] value. Let's evaluate each table one by one.
### Table 1
[tex]\[
\begin{tabular}{|c|c|}
\hline$x$ & $f(x)$ \\
\hline 5 & 26 \\
\hline 14 & 197 \\
\hline 7 & 50 \\
\hline 13 & 170 \\
\hline 12 & 197 \\
\hline
\end{tabular}
\][/tex]
In this table:
- [tex]\( f(5) = 26 \)[/tex]
- [tex]\( f(14) = 197 \)[/tex]
- [tex]\( f(7) = 50 \)[/tex]
- [tex]\( f(13) = 170 \)[/tex]
- [tex]\( f(12) = 197 \)[/tex]
We observe that both [tex]\( f(14) \)[/tex] and [tex]\( f(12) \)[/tex] map to the value 197. Therefore, this table does not represent a one-to-one function because the value 197 corresponds to more than one input ([tex]\( x \)[/tex] values 14 and 12).
### Table 2
[tex]\[
\begin{tabular}{|c|c|}
\hline$x$ & $f(x)$ \\
\hline-2 & 5 \\
\hline-1 & 2 \\
\hline 0 & 1 \\
\hline 1 & 2 \\
\hline 2 & 5 \\
\hline
\end{tabular}
\][/tex]
In this table:
- [tex]\( f(-2) = 5 \)[/tex]
- [tex]\( f(-1) = 2 \)[/tex]
- [tex]\( f(0) = 1 \)[/tex]
- [tex]\( f(1) = 2 \)[/tex]
- [tex]\( f(2) = 5 \)[/tex]
We observe that both [tex]\( f(-2) \)[/tex] and [tex]\( f(2) \)[/tex] map to the value 5, and both [tex]\( f(-1) \)[/tex] and [tex]\( f(1) \)[/tex] map to the value 2. Therefore, this table does not represent a one-to-one function because the values 5 and 2 correspond to more than one input.
### Table 3
[tex]\[
\begin{tabular}{|c|c|}
\hline$x$ & $f(x)$ \\
\hline 7.25 & 11 \\
\hline 7.5 & 13 \\
\hline
\end{tabular}
\][/tex]
In this table:
- [tex]\( f(7.25) = 11 \)[/tex]
- [tex]\( f(7.5) = 13 \)[/tex]
We observe that each [tex]\( f(x) \)[/tex] value is unique and none of the [tex]\( x \)[/tex] values map to the same [tex]\( f(x) \)[/tex] value. Therefore, this table represents a one-to-one function.
### Conclusion
The only table that represents a one-to-one function is the third table.
