Sanderson M. Smith
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COMPOSITION OF FUNCTIONS: A FINANCIAL EXAMPLE (A note sent to my Algebra II students)
The composition of functions is an important topic. It is often helpful to think of a function as a rule. The composition of functions consists of applying one rule, getting a result, and then applying the second rule to what you obtained from the first rule.
Here is a composition example relating to finance.
A store selling very expensive items will, on February 8, sell any item for $50 less than the listed price. On any day in February, the store will give a discount of 15% to any customer who can prove that he/she contributed to a local charity.
OK, let x be the listed price of an item in the store.
If P(x) is the price you will pay for an item on Feb. 8, then
P(x) = x - 50.
If D(x) is a price discounted at 15% , then, in February, the amount you will pay is
D(x) = 0.85x
This should make sense, folks. You have MATH POWER. You have two functions, P and D, which depend on the value of x, the listed price. If you don't know x, you can't calculate P(x) or D(x). And, note that if you do know x and "follow the rules," you get one value for P(x), and one value for D(x). That is, P and D are truly functions. P is a rule that says "take the listed price, than subtract $50." D is a rule that says "take the listed price, then take 85% of it."
Now, on February 8, my buddy, HERKIMER, who has a receipt to prove he has donated to a local charity, plans to make a purchase in this store. HERKY, who does have MATH POWER, realizes that a composition of functions is involved here, and he wonders if there is a difference between P(D(x)) and D(P(x)).
OK, folks... WHAT IS HERKY thinking about here?
You should be able to put this in words. (If you can't appreciate the question he is pondering, I hope that you find someone reliable to handle your finances in the future.)
Let's do some computations. Remember, P(x) = x - 50 and D(x) = 0.85x.
P(D(x)) = P(0.85x) = 0.85x - 50
D(P(x)) = D(x-50) = 0.85(x-50) = 0.85x - 42.50
If HERKY had a choice of order, which would he prefer?
You can, I hope, look at this and realize that there is a difference of $7.50, regardless of the price. In other words, the order in which the rules are applied does make a difference. OK, let's accept the fact that we are probably not going to have a heart attack over a difference $7.50, but that misses the point entirely. This is just one simple problem. In the real world, decisions are made in which the order things are done can make a difference of hundreds, thousands, or perhaps millions of dollars.
In this simple problem, if a customer enters the store on Feb. 8 with evidence that he/she has donated to a local charity, which is better? P(D(x)) or D(P(x))? You realize, of course, that this depends upon who is asking the question... the customer or the store manager!
The store manager would, of course, prefer D(P(x)). If the rules were applied in this order, then the composition rule D(P(x)) could be replaced by a simple rule, say W, that says
D(P(x)) = W(x) = 0.85x - 42.50.
In other words, the simplified rule W would say "take 85% of the listed price, then subtract $42.50."
Also, from the standpoint of the store, suppose on Feb. 8, one thousand items are purchased by people who can prove they donated to a local charity. Can you use your MATH POWER and determine that the order of the rules makes a difference of $7.50x(1,000) = $7,500 to the store?
You are probably familiar with the saying
"A fool and his money are soon parted."
This, of course, begs the question:
How did the fool and his money get together in the first place?
MATH POWER TO ALL.
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