Composition of Functions
Symbolic Composition
Application
You work forty hours a week at a furniture store. You receive a $220 weekly salary, plus a 3% commission on sales over $5000. Assume that you sell enough this week to get the commission.
Given the function f(x) = 0.03x and g(x) = x – 5000, which of (f o g)(x) and (g o f)(x) represents your commission?
Solution:
Using (f o g)(x) = f(g(x)) means subtract the sales x from $5000 that didn't earn commission, and then multiplying by 3%.
On the other hand, (g o f)(x) = g(f(x)) means multiply the sales x by 3%, and then subtract $5000 from the result. This could result in a negative number
Therefore, (f o g)(x) represents the commission.
Check the composite function with numbers that make sense in the situation. In the case of the commission function above, you could test the following sales values:
|
total sales |
$3000 |
$6000 |
$8000 |
|
commission |
$3000 – $5000 |
$6000 – $5000 |
$8000 – $5000 |
|
commission |
$0 |
(0.03)($1000) |
(0.03)($3000) |
For each sales value, subtract $5000 to see if there is a commission. Then multiply by 3%. The composite function would be f(g(x)) = 0.03(x - 5000).