Application of Composition of Functions: Real world examples? 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!

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 ...  


