Answer: Choice C
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Step-by-step explanation:
If you work 1 hour, then you earn 24*1 = 24 dollars
If you work 2 hours, then you earn 24*2 = 48 dollars
and so on
If you work x hours, where x is between 0 and 38, then you earn 24*x dollars
Note how I gave the restriction "where x is between 0 and 38". So the equation y = 24x only applies if x is between 0 and 38. Otherwise, we have to use another equation. So that explains why the top part of the piecewise function is 24x, as choice A through D shows. Unfortunately at this point we aren't able to eliminate any non-answers.
However, the second part isn't too bad.
If you work more than 38 hours, then another equation will apply. It states that you earn "$31 an hour for each hour worked over 38 hours". What does this mean? It is basically like an overtime structure. If you work past 38 hours, then you get paid 7 dollars more. It's not "time and a half" (1.5) like usual overtime, but it's at least something.
The second part will NOT be y = 31x. Why not? Because that implies that you get paid $31 for ALL hours instead of just the extra hours past 38 hrs. How do we account for just the extra hours? By using subtraction. Instead of just 'x', we'll have 'x-38'. For example, if you worked 40 hours then you worked the first 38 hrs under $24 an hour and then an additional 2 hours for the overtime pay.
If x = 40, then x-38 = 40-38 = 2 shows algebraically how to get that value.
So 31(x-38) properly calculates the extra amount of money earned for the overtime. Keyword: extra. The previous amount of money earned under the first equation must be considered. If you worked 38 hours for $24 and hour, then you earn 24*38 = 912 dollars. This is the most you can earn before overtime kicks in. After overtime starts, that's when you'll add on 31(x-38). Overall, the second equation is y = 31(x-38)+912 once you put all the pieces together
So to recap, the first piece is
y = 24x
and the second piece is
y = 31(x-38)+912
The first piece only applies if x is 0, or x is 38 or if x is in between those values
The second piece only applies if x is larger than 38
This helps us find the amount of money earned (y) for the number of hours worked (x). This piecewise function has 24x as the standard wage pay for the first 38 hours, then overtime pay for any additional hours over 38 hrs. This is why choice C is the answer.
Note: the piecewise notation used in choice C is basically two equations combined, but they switch depending on what x is equal to. It might be confusing at first since the notation isn't exactly friendly. I suggest breaking things up if it gets too tricky.
Extra Stuff: The graph is attached. It's probably not needed, but it might be handy somehow. The red portion is the first part y = 24x (from x = 0 to x = 38) and the blue part is y = 31(x-38)+912 for any x > 38. Take note how the graph is slightly steeper after point B, which is to reflect the higher wage after overtime kicks in.