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Looking at Inverse Functions in a Real-World Context

Waterloo Park posted the following schedule listing the number of hours an employee works on a given day. Let [tex]B(x)[/tex], [tex]T(x)[/tex], [tex]R(x)[/tex], and [tex]S(x)[/tex] represent the number of hours worked by Bill, Ted, Rufus, and Socrates, respectively, on a given day [tex]x[/tex].

\begin{tabular}{|l|r|r|r|r|r|}
\hline & Mon & Tues & Wed & Thurs & Fri \\
\hline Bill & 6 & 5 & 4 & 7 & 8 \\
\hline Ted & 4 & 0 & 6 & 10 & 3 \\
\hline Rufus & 8 & 3 & 6 & 5 & 0 \\
\hline Socrates & 1 & 2 & 3 & 4 & 0 \\
\hline
\end{tabular}

What is the value of the inverse shown below?

[tex]\[S^{-1}(0) = \square\][/tex]



Answer :

GinnyAnswer
To understand the given problem, let us look at the schedule provided by Waterloo Park that depicts the number of hours each employee works from Monday to Friday. Specifically, we'll focus on Socrates' work schedule, represented by [tex]\( S(x) \)[/tex].

Here is the data for Socrates:
[tex]\[ \begin{tabular}{|l|r|r|r|r|r|} \hline & Mon & Tues & Wed & Thurs & Fri \\ \hline Socrates & 1 & 2 & 3 & 4 & 0 \\ \hline \end{tabular} \][/tex]

The question asks for [tex]\( S^{-1}(0) \)[/tex]. This notation represents the inverse function value, which essentially means we are looking for the day [tex]\( x \)[/tex] when Socrates works 0 hours.

To find this, we'll inspect each day and check when [tex]\( S(x) = 0 \)[/tex]:
- On Monday (Mon), [tex]\( S(Mon) = 1 \)[/tex]
- On Tuesday (Tues), [tex]\( S(Tues) = 2 \)[/tex]
- On Wednesday (Wed), [tex]\( S(Wed) = 3 \)[/tex]
- On Thursday (Thurs), [tex]\( S(Thurs) = 4 \)[/tex]
- On Friday (Fri), [tex]\( S(Fri) = 0 \)[/tex]

From the above, we can see that the only day when Socrates works 0 hours is Friday.

Therefore, the value for [tex]\( S^{-1}(0) \)[/tex] is:

[tex]\[ S^{-1}(0) = \text{Fri} \][/tex]

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