After a weekend away from composite functions, today’s opener was designed to bring functions back into discussion, disguised as an innocent-looking shopping problem:

**It’s the day of the big department store sale, and you have two coupons you have clipped from the newspaper. One coupon takes $10 off the price of any item, while the other takes 30% off the price. In what order should these discounts be taken for you to realize the maximum savings?**

After a few minutes of table talk, just about all groups agreed that taking the 30% off first would seem an optimal strategy. But when asked to provide justification, groups took much different paths.

Some felt choosing a dollar value would provide adequete justification:

How many values are needed to convince ourselves that this strategy is optimal? Is it possible that one strategy is best for some prices, while the other is best for others?

Another group shared the “I know I am right…just because” method

Not very elegant…nor very convincing. But a ray of sunshine appears from the other side of the room, as a group considers defining functions to represent the discounts….but stops just short of pursuing them as a proof.

The eventual “proof” done via composite functions shows that not only is one method superior – it will always be superior by 3 dollars. Add in a domain restriction that our starting value must be at least 10 dollars, and we have successfully reviewed all of our scary function vocabulary.

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Greetings!

Today’s silly questions that I am sure you must have answered previously: Is there a protective covering/coating on the desktops (and if so, what is it), what kind of markers did you use (dry erase,wet erase, etc.), and what do you use to erase/remove the markings on the desk (windex, water, etc.).

THANKS!

Sincerely,

John J. Kalicki

I don’t do anything special to the desks. Whatever surface they are made of works nicely with the markers, and I keep a bottle of 409 handy in the room. I’ve been doing it for years and have never had gruff from admins or custodians.

Nice. I need to review functions for the final!

I do this same thing (but don’t use the desks).

Tell them to walk into Kohl’s with both coupons. They decide which coupon must go first, so then you’ll know you get the biggest discount the other way. (hehe)

I would take it one step further – the $10 spending restriction would apply only to the $10 coupon. In terms of formulation, we would need to probably define one function piece-wise in order to align domains and ranges (savings for $10 coupon is $0 if the amount owed is less than $10). Further, this would create a situation where $10 coupon first would be beneficial in certain cases.

A question that came to my mind: At what price(s) would the item be free?

At what prices can you not use both coupons?

Really enjoyed this question and this post. Thanks again!