Question

Let f : A → B, g : B → C, and h : C → D be functions.

1. State what you need to show to conclude that h ◦ (g ◦ f) = (h ◦ g) ◦ f. 13

2. Consider now some a ∈ A. Calculate h((g ◦ f)(a)) and (h ◦ g)(f(a)). Are they equal?

3. Use your solutions to (1)–(2) to conclude that h ◦ (g ◦ f) = (h ◦ g) ◦ f.

Answer 1

1. The definition of compositions shows that both compositions $h \circ(g\circ f)$ and $(h \circ g) \circ f$ are functions.

2.

$$h \circ(g\circ f)(a)=h(g\circ f(a))=h(g(f(a)))$$

$$=h \circ g( f(a)=(h \circ g) \circ f(a)$$

3. The composition of functions is always associative. That is, if $f, g,$ and $h$  are composable, then $h \circ(g\circ f) = (h \circ g) \circ f.$