I've recently encountered on MSE the claim as to :
$\operatorname{dom}(g \circ h) = \operatorname{dom}(h) \cap h^{-1}(\operatorname{dom}h)$.
If I am correct, this reads, in words as :
the domain of the function composition $g \circ h$ is the intersection of the domain of $h$ and of the image of the domain of $h$ under the inverse function of $h$.
Is this correct?
How to explain where the inverse function comes from?
$h$ needn't have an inverse. The notation $h^{-1}(A)$ means the set of all $x \in \textrm{dom}(h)$ such that $h(x) \in A$. Because $h^{-1}(A) \subseteq \textrm{dom}(h)$, we can say that $\textrm{dom}(g \circ h) = h^{-1}(\textrm{dom}(g))$. In other words, in order for $g(h(x))$ to be defined, $x$ must be in the domain of $h$, and $h(x)$ must be in the domain of $g$.