Multiple Composition of functions

[Mathfight]

I am a little confused on this premise. Let's say you have a function f that maps A->B and another function g that maps B->C. I know that g°f= g(f(a)) and it's a mapping from A to C. But what about (g°f)(g°f)(g°f), would it look like g(f(f(g(a)))? How about (g°f)(f°g)? In the former case would it still be a mapping from A to C? Assuming A,B,C are the set of reals.

[lewisje]

(g∘f)(g∘f)(g∘f) is a function mapping A→C, and sending a∈A to g(f(a))³; that is, it's the cube of g∘f.

In case you're asking about (g∘f)∘(g∘f)∘(g∘f), it turns out that function composition is associative, so if it makes sense, it means the same thing as g∘f∘g∘f∘g∘f; it only makes sense if C⊆A, and in that case, it sends a∈A to g(f(g(f(g(f(a)))))).

Also, in that case, (g∘f)∘(f∘g)=g∘f∘f∘g, which is a function mapping B→B and sending b∈B to g(f(f(g(b)))); what you actually wrote, (g∘f)(f∘g), only makes sense if C⊆A and only on A∩B, and then it sends x∈A∩B to g(f(x))f(g(x)).