Function Composition in Word Problems

We often do this when we're working vertically, such as the following:

Strictly speaking, the above was correct until I did the subtraction; 120×27 (at the beginning) is not equal to 1840 (at the end). I should have done the subtraction as its own step:

If you've done the first sort of computation and your instructor said that this wasn't right, that you had to show each step separately— Well, your instructor was correct, in the sense that you were putting "equals" signs between things that weren't actually equal. But, if you think about it a different way, you were kind-of composing functions.

How so? If we look at the computations as a sequence of operations to be applied to some given input, then the above could be viewed as f (x) = 27x and g(x) = x − 1400, with the input value being x = 120. And the sequence of steps could have been done as a function composition; namely, as (g ∘ f )(120)

Does anyone do function composition in real life?

Composition of functions, as a process at least, is used by people every day in real life. Composition of functions allows us to do many computations in a row; when working on real-life things like taxes, one often does many computations in a row. Thus, many of the steps for filing one's taxes may be viewed as representing the composition of functions. For instance, instructions for some section may say something like this:

Take the value from line 31. Multiply by 0.03. Subtract 10,500 from this value. Write down this amount or, if this amount is less than zero, write down zero.

If we label the original "value from line 31" as x, then the multiplication could be viewed as being f (x) = 0.03x. The subtracting could be g(x) = x − 10,500. Then the last step would be something like h(x) = max(x, 0). The tax computation, from beginning to end, could be viewed as the following composition:

So composition of functions can be implicit in everyday life. As a result, you may be given "real life" word problems involving composition. For these exercises, you'll need to think about what's going on, the order in which things are being done, and therefore the way in which the modelling functions need to be composed.

What is an example of a word problem that uses function composition?

• 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 ∘ g)(x) and (g ∘ f )(x) represents your 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 represent the same process as whatever you did. In the case of the commission formula above, you could test the following sales values (all above the minimum for commissions):

• total sales: $6000
• commission sales: $6000 − $5000 = $1000
• commission: 0.03($1000) = $30

• total sales: $8000
• commission sales: $8000 − $5000 = $3000
• commission: 0.03($3000) = $90

• total sales: $15000
• commission sales: $15000 − $5000 = $10000
• commission: 0.03($10000) = $300

For each sales value, I first subtracted $5000 to see how much was getting a commission. Then I multiplied this amount by 3% to find out how much commission I was getting. By looking at this pattern, I could see that I should apply the "subtract five thousand" formula first, and apply the "multiply by three percent" formula last. This matches f (g(x)) = (f ∘ 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.
  1. 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.
  2. Calculate and interpret (f ∘ t)(x) and (t ∘ f )(x). Which results in a lower cost to you?
  3. Suppose taxes, by law, are not to be charged on delivery fees (which represent a "service"). Which composite function must then be used?

---

• 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.