I've taught function composition in Algebra II and in Pre-Calculus, and my students ALWAYS seem to struggle with it. I remember it actually bringing a girl to tears in one of my first years of teaching. So this year I decided to try and try a more informal approach to introduce the concept in hopes that it would make something "click" for my students.
I printed this out and cut apart the functions.
Function Composition Relay
Each person in a group got a different function. Then I wrote on the board the definitions of composition (f (g(x)) and g(f(x))) and explained that it was like a relay, with the inside handing off to the outside. I started with an example of composition on a number: f(g(2)) and explained that this meant that g would take the 2 and do "its thing" to the 2, then hand off the result to f and then f would do "its thing" and get the final result. I posted several more examples of composition on a number on the board, one at a time, and had them calculate the results in their group. We even did composition of three and even four functions. Then I asked them "What if I want to know a formula for the relay so that I don't have to actually do the relay in order to figure out how it turns out? In other words, what if I want to find the composition on x?" We then did f(g(x)) together. They were able to do any combination of function composition I put on the board, including (with a little help) the f(g(h(j(x)) composition. It went great! I will definitely be teaching composition as a relay from now on.
I will say that my students still struggled to recognize the symbol for composition when it showed up on the test. If I reminded them that it was the "relay" operation, they would go "oh, yeah!", so I'm going to have to give them some more independent practice with the notation next year....