Composition of Functions: Word Problems using Composition


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Composition of Functions:
Word Problems using Composition (page 4 of 5)

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 problems where you need to think about what you're doing, the order in which you're doing it, and therefore the way in which 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 territory! (Would I owe money to my boss then?)

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

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 purchase amount x. Write another function f(x) for the total, including the delivery fee, on 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). That is, if I have to pay taxes on the purchase plus the delivery fee, I would pay more. I would rather 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, as a function of time, is: Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

r(t) = 2.5t

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