Going back to function machines to help pupils come to grips with the idea of composite functions. I find that starting with numerical examples is better than heading straight for ‘x’. Specific before general is often a good rule of thumb in explaining ideas to pupils.

I also ask pupils to come up with a couple of functions of their own and try chaining them together, substituting different values each time. It’s also nice if you offer them some functions which are inverse of each other. This representation really captures the idea that g(g-1(x)) = x etc.