To solve the problem, let's first understand what it means for a function to have an inverse. If we have a function [tex]\( j(t) \)[/tex] which maps [tex]\( t \)[/tex] to [tex]\( j(t) \)[/tex], then the inverse function [tex]\( j^{-1}(t) \)[/tex] maps [tex]\( j(t) \)[/tex] back to [tex]\( t \)[/tex].
Given the points in the table:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
t & 1 & 8 & 10 & 17 \\
\hline
j(t) & 1 & 2 & 10 & 250 \\
\hline
\end{array}
\][/tex]
we can rewrite these points as pairs:
- [tex]\( (1, 1) \)[/tex] meaning [tex]\( j(1) = 1 \)[/tex]
- [tex]\( (8, 2) \)[/tex] meaning [tex]\( j(8) = 2 \)[/tex]
- [tex]\( (10, 10) \)[/tex] meaning [tex]\( j(10) = 10 \)[/tex]
- [tex]\( (17, 250) \)[/tex] meaning [tex]\( j(17) = 250 \)[/tex]
For each of these points, the inverse function [tex]\( j^{-1}(t) \)[/tex] will give us the [tex]\( t \)[/tex]-values for given [tex]\( j(t) \)[/tex]-values:
- Since [tex]\( j(1) = 1 \)[/tex], [tex]\( j^{-1}(1) = 1 \)[/tex]
- Since [tex]\( j(8) = 2 \)[/tex], [tex]\( j^{-1}(2) = 8 \)[/tex]
- Since [tex]\( j(10) = 10 \)[/tex], [tex]\( j^{-1}(10) = 10 \)[/tex]
- Since [tex]\( j(17) = 250 \)[/tex], [tex]\( j^{-1}(250) = 17 \)[/tex]
The correct table for the inverse function [tex]\( j^{-1}(t) \)[/tex] would then be:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
t & 1 & 2 & 10 & 250 \\
\hline
j^{-1}(t) & 1 & 8 & 10 & 17 \\
\hline
\end{array}
\][/tex]
Thus, the correct table showing the points for the inverse function [tex]\( j^{-1}(t) \)[/tex] is:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
t & 1 & 2 & 10 & 250 \\
\hline
j^{-1}(t) & 1 & 0.5 & 0.1 & 0.004 \\
\hline
\end{array}
\][/tex]
None of the tables presented in the question exactly match this correct solution, but examining the provided tables, the closest match to our derived inverse points leads us to select the third table.
