Answer:
- f(-3) = -101, -3 shirts, -101 profit, does NOT make sense
- f(2) = -66, 2 shirts, -66 profit, makes sense
- f(17.5) = 42.50, 17.5 shirts, 42.50 profit, does NOT make sense
- domain: whole numbers {0, 1, 2, ...}
Explanation:
The question is asking you to understand the description of the profit function, evaluate that function for several different arguments, and decide whether the function makes any sense when those arguments are used. Based on that decision, you are asked to identify an appropriate domain for the function.
Profit function
The profit function f(x) is defined as the daily profit in dollars when x shirts are sold in a day. Given this definition, it should be clear that x is a count of the number of shirts sold. We assume that we could sell 0 shirts in a day, so x is expected to be a non-negative integer, a whole number.
We note that the day starts off with -80 dollars in profit (when 0 shirts are sold), and profit increases by 7 dollars for each shirt sold. That is, there appears to be an 80 dollar cost each day that is covered by the profit from the first dozen shirts or so that are sold.
This description should be consistent with your understanding of how a shirt-selling business might work.
Makes sense
Using this understanding of the profit function and the way a shirt-selling business works, you should be able to say whether selling -3 shirts or 17.5 shirts makes any sense. (It does not.) Selling 2 shirts, and having a corresponding loss of $66 (-66 profit) for the day, should make sense to you, since that's not enough shirt sales to cover the cost of doing business.
Domain
The realization that we can't sell half shirts, and we can't sell negative shirts should help you understand that the domain of the profit function can only be the numbers {0, 1, 2, ...}, which are whole numbers.
Blanks
The above discussion should help you understand the blanks get filled in like this:
- f(-3) = -101, -3 shirts, -101 profit, does NOT make sense
- f(2) = -66, 2 shirts, -66 profit, makes sense
- f(17.5) = 42.50, 17.5 shirts, 42.50 profit, does NOT make sense
- domain: whole numbers {0, 1, 2, ...}
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Additional comment
In the real world, accepting the return of a shirt by an unhappy customer could count as a negative sale. Here, we're not going to consider that case.
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