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!

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.
