Piecewise linear functions

Question

Derek earns $25 per hour (or part of an hour) for a regular 40-hour work week. Sometimes, he works over 40 hours a week and earns “double time” for the extra hours worked. (That means that his hourly rate for the additional hours over 40 hours per week is double his normal hourly rate.) Per work regulations, Derek can work at most 50 hours a week. 

A) If Derek works 46 hours in one week, how much would he earn that week? (Make sure you include units)

B) Create a piecewise linear function to represent the amount Derek will earn in one week if he works xxhours. (Make sure to define your variables and use proper notation)

C) Graph this function in an appropriate window. (Make sure to label your axes, scale and any important points.) 

D) Is your graph continuous?

Answer 1

To begin looking for the answer to this question, we will need to look at each component. In the question we will use variable 'x' for hours. Breaking apart the question, we have 2 parts.

1) Derek works between 0 and 40 hours

2) Derek works over 40 hours.

Therefore, your piecewise function should be composed of two linear equations where f(x) is a function of dollars.

Let's look at first one. Between 0-40 hours, Derek earns $25 per hour. To get the units of $, hours has to be multiplied to the amount per hour.

f(x) = 25x [ Amount of dollars = ($25/hour) x (hours)

Second portion is when Derek worked more than 40 hours. Same method applies for the first section.

f(x) = 2*25x or f(x) = 50x

However, if it is greater than 40, we have to account for the first 40 hours worked. Therefore, we include a -1000 at the end which is because of 25*40. This makes second equation:

f(x) = 50x - 1000

From here we have

f(x) = { 25x 0<=x<=40}

{ 50x - 1000 x>40}

Using this, we will begin answering the question.

1) We have x = 46. Therefore we use the second equation

f(x) = 50x-1000

f(x) = 50(46) - 1000

f(x) = 2300 - 1000

f(x) = 1300

2) See the step above.

f(x) = { 25x 0<=x<=40 }

{ 50x - 1000 x>40 }

3)

4) Yes.