Brenton's weekly pay, [tex]$P(h)$[/tex], in dollars, is a function of the number of hours he works, [tex]$h$[/tex]. He gets paid $20 per hour for the first 40 hours he works in a week. For any hours above that, he is paid overtime at $30 per hour. He is not permitted to work more than 60 hours in a week.
Which set describes the domain of [tex]$P(h)$[/tex]?
A. {h | 0 ≤ h ≤ 40}
B. {h | 0 ≤ h ≤ 60}
C. {P(h) | 0 ≤ P(h) ≤ 1,400}
D. {P(h) | 0 ≤ P(h) ≤ 1,800}
Answer (2)
The domain of a function represents the set of all possible input values.
Brenton can work between 0 and 60 hours per week, inclusive.
The domain of P ( h ) is the set of all h such that 0 ≤ h ≤ 60 .
The correct answer is { h ∣ 0 ≤ h ≤ 60 } .
Explanation
Understanding the Problem The problem asks for the domain of the function P ( h ) , which represents Brenton's weekly pay as a function of the number of hours he works, h . The domain consists of all possible values of h .
Determining the Bounds of h Brenton can work a minimum of 0 hours and a maximum of 60 hours per week. This is because he is not permitted to work more than 60 hours in a week. Therefore, the number of hours he works, h , must be between 0 and 60, inclusive.
Defining the Domain The domain of P ( h ) is the set of all possible values of h , which are the number of hours Brenton can work in a week. Since h can be any value between 0 and 60, inclusive, the domain is described by the inequality 0 { h ∣ 0 ≤ h ≤ 60 } .
Final Answer The correct set describing the domain of P ( h ) is the set of all h such that 0 ≤ h ≤ 60 .
Examples
Understanding the domain of a function is crucial in many real-life scenarios. For example, if you are planning a road trip, the domain could represent the possible number of hours you can drive each day. Knowing the domain helps you determine the total distance you can cover within a specific timeframe. Similarly, in a manufacturing process, the domain might represent the number of units you can produce given certain resource constraints. By understanding these limitations, you can make informed decisions and optimize your plans.
Brenton's weekly pay function P ( h ) has a domain that includes all possible hours he can work within a week, which ranges from 0 to 60 hours. Therefore, the correct answer is { h ∣ 0 ≤ h ≤ 60 } .
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