Real World Piecewise Functions

Up until recently, I never wondered when one uses piecewise funcitons in real life.  It wasn't important to me because I was only familiar with the mathematics.

After creating the hands on activity, it hit me, I didn't know when the piecewise function was used in real life.  There has to be some application because it exists since mathematics often explains the world.

So after a bit of research, I discovered several real world situations that are not contrived and make sense to me.

I assume most people reading this column have at some point worked a job where you were paid by the hour.  I had one where I worked filling the newspapers with sales flyers so I'd have days free to substitute and look for a teaching job.  The place I worked set the pay with the following parameters:  I was paid a per hour rate for the first 8 hours worked in a 24 hour period from midnight to midnight.  Anything over the 8 hours and I received time and a half for those hours.  If I worked holidays such as Christmas, I received double time.  That makes my pay a piecewise function because the amount I received depending on the number of hours I worked.

For many other jobs the piecewise is based on working a 40 hour week before a person begins to receive time and a half for overtime or double time for certain holidays or Sunday.  It all depends on the rules of the business.

Another piecewise function is based on the amount of something purchased.  The more of the one item you purchase, the less per item you pay.  I've dealt with a company where I paid full price for the first 10 items, the next 20 granted a 10 percent discount, etc.  I've seen this type of arrangement with t-shirts, water bottles, etc. 

Look at the launching of a rocket and follow its acceleration from launch to touchdown to see a piece wise function.  For the first two seconds, the rocket accelerates from zero to a certain speed.  Between two and twelve seconds, the rocket slows down as it begins to coast and due to gravity.  After 12 seconds, the parachute is released and the rocket floats back to earth. Based on the above situation, I think anyone who jumps out of a plane is going one velocity until they pull the ripcord causing the parachute to open and they are now floating down.

The next time I teach this topic, I can provide examples without students having to ask.  I feel like I've scored a win. 

Let me know what you think.  I love to hear from people.  Thanks for reading.