Composition of Functions:
     Word Problems using Composition (page 5 of 6)

Sections: Composing functions that are sets of point, Composing functions at points, Composing functions with other functions, Word problems using composition, Inverse functions and composition

You may be given exercises where you need to think about what's going on, the order in which things are being done, and therefore the way in which modelling functions need to be composed:

• You work forty hours a week at a furniture store. You receive a $220 weekly salary, plus a 3% commision on sales over $5000. Assume that you sell enough this week to get the commission. Given the functions f (x) = 0.03x and g(x) = x – 5000, which of ( f o g)(x) and (g o f )(x) represents your commission?

Well, ( f o g)(x) =  f(g(x)) would mean that I would take my sales x, subtract off the $5000 that didn't get the commission, and then multiply by 3%. On the other hand, (g o f )(x) =
g( f (x)) would mean that I would take my sales x, multiply by 3%, and then subtract $5000 from the result. This could land me in negative numbers! (Would I owe money to my boss?)

So ( f o g)(x) does what we need it to do: ( f o g)(x) represents my commission.

If you're not sure how the formulas are working, try plugging in numbers that you can understand, and pay attention to what you do with those numbers. The formula you need will be the same process. In the case of the commission formula above, you could test the following sales values:

For each sales value, I first subtracted $5000 to see if I'd sold enough to earn any commission at all. If I had, then I multiplied by 3%. Then I should apply the "subtract five thousand" formula first, and then apply the "multiply by three percent" formula last. This matches f(g(x)) = ( f o g)(x), which confirms my earlier answer.

• You make a purchase at a local hardware store, but what you've bought is too big to take home in your car. For a small fee, you arrange to have the hardware store deliver your purchase for you. You pay for your purchase, plus the sales taxes, plus the fee. The taxes are 7.5% and the fee is $20. 
  (i)  Write a function t(x) for the total, after taxes, on the purchase amount x. Write another function f(x) for the total, including the delivery fee, on the purchase amount x.
  (ii)  Calculate and interpret ( f o t)(x) and (t o f )(x). Which results in a lower cost to you?
  (iii)  Suppose taxes, by law, are not to be charged on delivery fees. Which composite function must then be used?

This sort of calculation actually comes up in "real life", and is used for programming the cash registers. And this is why there is a separate button on the register for delivery fees and why they're not rung up as just another purchase.

(i)  The taxes are 7.5%, so the tax function is given by t(x) = 1.075x

The delivery fee is fixed, so the purchase amount is irrelevant.

The fee function is given by f (x) = x + 20

(ii)  Composing, I get this:

( f o t)(x) = f (t(x)) = f (1.075x) = 1.075x + 20
(t o f )(x) = t( f (x)) = t(x + 20) = 1.075(x + 20)
    = 1.075x + 21.50

Then I would pay more using (t o f )(x), because I would be paying taxes (the t(x) formula) on the delivery fee (the "+20" in the f (x) formula). I would prefer that the delivery fee be tacked on after the taxes, because ( f o t)(x) results in a lower cost to me.

(iii)  If the state is not allowed to collect taxes on delivery fees, then:

The function to use is ( f o t)(x).

• Your computer's screen saver is an expanding circle. The circle starts as a dot in the middle of the screen and expands outward, changing colors as it grows. With a twenty-one inch screen, you have a viewing area with a 10-inch radius (measured from the center diagonally down to a corner). The circle reaches the corners in four seconds. Express the area of the circle (discounting the area cut off by the edges of the viewing area) as a function of time t in seconds.

Since the circle's leading edge covers ten inches in four seconds, the radius is growing at a rate of (10 inches)/(4 seconds) = 2.5 inches per second. Then the equation of the radius r, as a function of time t, is:   Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

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r(t) = 2.5t

The formula for the area A of a circle, as a function of the radius r, is given by:

Then the area, as a function of time, is found by plugging the radius equation into the area equation, and simplifying the composition:




Then the function they're looking for is:

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