How to write a piecewise function

Question

You have a summer job that pays time and a half for overtime. That is, you work more than 40 hours a week, then you are paid 1.5 times your normal hourly wage of $8 for each hour of overtime. 

 

Write a piecewise function that related hour worked to total wages. 

Answer 1

The function (I'll call it W(x), for total wages, given x hours worked.) given has two pieces:

 

1. Hours worked is less than or equal to 40 (no overtime).

2. Hours worked is more than 40 (overtime).

 

Translate these into mathematical conditions:

 

1. x ≤ 40

2. x > 40

 

The pay (W(x)), in dollars, while a given condition is true (the value of x is in that "box"):

 

1. W(x) = 8x

2. W(x) = 320 + 12(x - 40)

 

How did I get #2? The 320 is from the first 40 hours of work. At x = 40, W(x) will yield 320. The 12 is from 1.5 × $8 (the normal rate). The (x - 40) is "for every hour over 40".

 

Put it all together:

 

W(x) = 8x, when x ≤ 40.

            320 + 12(x - 40), when x > 40.

 

If you graph this function, it will look like a bent line.