Application of Composition of Functions: Real world examples?

Question

Do you know of a real world example where you'd combine two functions into a composite function? I see this topic in Algebra 2 textbooks, but rarely see actual applications of it. It's usually plug and chug where you take f(g(4) and run it through both functions. This leads to the idea of creating a composite function f(g(x). But it's somewhat academic, and it's not like you're saving time b/c you need to run 50 different numbers through both functions.

While on this topic, where is this topic used in later math? In Precalculus, you can determine the domain of the composite function. In Calc, composition is used to describe the ideas behind the Chain Rule. In Calc, you break down a function into the 2 components to show it's continuous. (If the components are continuous, so is the composite function) Any other main areas?

Thanks!

Answer 1

First example of Algorythms: You have a list, compose by a head (an element) and a tail (a list). A composition of functions could return the second element of the list, let's say, L:

$ Head(Tail(L)) $

This is a simple examen in my field of study, I don't know if that's what you're looking for.

Answer 2

Let $A$ be the set of fruit in your preferred grocery store. There is a function $f:A\to \mathbb R^+$ that maps a fruit to its weight in kilogramss. There is also a function $g:\mathbb R^+\to \mathbb R^+$ which converts a kilogram weight to the same weight measured in tonnes.

Consider the function $g\circ f:A \to \mathbb R$. It is the function which measures the weight of the fruit of your grocer in tonnes.

Answer 3

Think of an industrial plant that produce bottles of beer; first there is the operation (or function) $f_1$ that puts the beer inside the bootle, followed by the opeartion $f_2$ that close the bottle with the cap.

Clearly, these two functions (operations) do not commute ...

Answer 4

One common everyday life example is unit conversion. If I know how many miles and hours I have been walking, but want to decide what my average speed is in feet per second, generally what ends up happening is I combine several functions to get my result. I know the number of feet in a mile, thus if $x$ is in miles per unit time, I have a function $f(x)$ which converts it to feet per unit time. I also have $g(x)$ which converts length per unit hours to length per unit seconds (since I know there are $60$ seconds in a minute). I can use composition of functions to take my miles per hour speed and convert it to feet per second; if I am doing this computation often enough, it will be helpful for me to memorize the composed function $h(x) = g ( f(x))$.

Basically any time you are doing one computation, $f$, and then using that result to do another computation, $g$, that is composition of functions. In particular, if you wanted to make this computation go faster in the future, you could memorize the function $g \circ f$.

Some more real-word examples can be found here.

Answer 5

Many people believe that $+$ is a binary function. They are wrong. Since Haskell Curry it is known that $+$ is in fact an unary function, operating on numbers and returning another function, also operating on numbers. $a+b$ is a shorthand for a composition $(+(a))(b)$

Same goes for any other $n$-ary function, of course.

Answer 6

You use composite functions whenever you buy a sale (discounted) item. When you are standing in the store trying to decide if you can afford the item, the first thing you calculate is the discount. For example, I want to buy this 20 dollar shirt, and it is on sale at 15% off. This means that the shirt is really 17 dollars. Now, you must calculate what the shirt will cost after sales tax (let's say it is 8%). Your total cost for the shirt after the discount and sales tax will be $18.36. This process of computation can be expressed as a composite function.

If f(x) = The price of the shirt after the discount and g(x) = The price after sales tax then,

The function for the final cost of the shirt = g(f(x)).