Let's carefully analyze the problem given to understand how Brenton's weekly pay [tex]\( P(h) \)[/tex] relates to the number of hours [tex]\( h \)[/tex] he works and determine the correct domain of the function [tex]\( P(h) \)[/tex].
1. Understanding the Pay System:
- Brenton is paid [tex]$20 per hour for the first 40 hours.
- For any hours worked beyond 40 hours, he is paid $[/tex]30 per hour.
- He is allowed to work a maximum of 60 hours in a week.
2. Determining the Possible Values of [tex]\( h \)[/tex]:
- The lowest number of hours Brenton can work is 0 hours.
- The highest number of hours Brenton is permitted to work is 60 hours.
Therefore, [tex]\( h \)[/tex] can range from 0 to 60.
3. Formulating the Domain:
- The domain of [tex]\( P(h) \)[/tex] includes all possible values of [tex]\( h \)[/tex] that Brenton can work in a week.
- Since [tex]\( h \)[/tex] is the number of hours worked and we have established that [tex]\( h \)[/tex] can range from 0 to 60,
the domain is all integers from 0 to 60.
4. Identifying the Correct Domain:
- Let’s compare this with the options provided:
[tex]\[
\begin{aligned}
&\{h \mid 0 \leq h \leq 40 \} \quad \text{(This is incorrect because } h \text{ can go up to 60 hours)} \\
&\{h \mid 0 \leq h \leq 60 \} \quad \text{(This is correct as discussed above)} \\
&\{P(h) \mid 0 \leq P(h) \leq 1,400 \} \quad \text{(This describes the range of } P(h) \text{, not the domain)} \\
&\{P(h) \mid 0 \leq P(h) \leq 1,800 \} \quad \text{(This also describes the range of } P(h) \text{, not the domain)} \\
\end{aligned}
\][/tex]
Therefore, the set that describes the domain of [tex]\( P(h) \)[/tex] is [tex]\( \{h \mid 0 \leq h \leq 60 \} \)[/tex].
