Jack works for a company that pays him $20 an hour. his normal workday does not exceed 8 hours. after working 8 hours in a day, he gets paid $5 per hour overtime in addition to his regular wage. he cannot work fewer than 5 hours a day or more than 14 hours a day. which graph shows jack's wages if he works for x hours a day?

graph A (see picture attached)

This question is about understanding how a piecewise function works and the intervals in which it is defined. Indeed, all the graphs are similar, the only difference is the extremities (full or empty dots).

We know:
20 $/h = regular wage
x = number of hours worked
5 $/h = extra wage

We can say that Jack's regular wage can be defined by the function:
y₁ = 20 · x  
if 5 < x < 8
(later we will consider if the extremities of the interval are included or not)

We can also say that Jack's wage if he works overtime, can be defined by the function:
y₂ = 5 · (x - 8) + 160 
if 8 < x < 14
where: 
(x - 8) takes into account that the first 8 hours worked are paid with the regular wage,
160 is the regular wage earned in 8 hours.

We now need to understand what happens in the extremities of the intervals:
- Jack can work only 5 hours (not less), therefore 5 is included;
- Jack can work 14 hours (not more), therefore 14 is included;
- if Jack works exactly 8 hours, he gets paid according to the regular wage; this means that 8 is included in the first interval and excluded in the second one.

Hence, we need to look for the graph representing the intervals:
5 ≤ x ≤ 8
8 < x ≤ 14

Remembering that an included extremity is represented by a full dot, while an excluded extremity is represented by an empty dot, the correct graph is option A).

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