Inverse of inverse of function?

Question

What is the inverse of inverse of a function (I assume the original function is invertible)? Is this the original function? Is it always true?

Answer 1

Yes it is the original function. By definition the inverse of $f:X\rightarrow Y$ is (unique if it exist) the function $g:Y\rightarrow X$ such that $g\circ f:X\rightarrow X$ and $f \circ g:Y \rightarrow Y$ are the identities on $X$ and $Y$. With that I mean that $g\circ f(x)=x$ for all $x\in X$ and $f\circ g(y)=y$ for all $y\in Y$. Note that if $g$ is the inverse of $f$, then $f$ is the inverse of $g$ by symmetry of this definition.

Answer 2

Clearly, the set of all bijective functions $$f: X → X$$ form a group with respect to the composition operator, so your claim makes sense.