Let $A$ be a set and $f : A \to A$ be a function. Define
$$f^{(n)}=\overbrace{f\circ f \circ \cdots \circ f}^{\text{$n$ times}}$$
a. Find a sensible meaning for $f^{(0)}$
b. Prove or disprove: $(f\circ g)^{(2)} = f^{(2)}\circ g^{(2)}$
c. Prove or disprove: If $f$ is invertible, ${f^{-1}}^{(n)} = {(f^{(n)})}^{-1}$
Any suggestions? I don't really know where to start
Hint: by the definition you know that $f^{(n+1)} = f^{(n)}\circ f$.
For (a), we should have $f^{(0)} \circ f = f$. What function $A \to A$ doesn't do anything when composed with another function?
For (b), write our what $(f\circ g)^{(2)}$ and $f^{(2)}\circ g^{(2)}$ look like. Remember that in general composition of functions is not commutative.
For (c), recall that a function $g:A \to A$ is an inverse of $f:A \to A$ if and only if $f\circ g= \mathrm{id}_A$. Show that $(f^{-1})^{(n)} \circ f^{(n)} = \mathrm{id}_A$, so $(f^{-1})^{(n)} = (f^{(n)})^{-1}$.