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.