1. True - Because if a relation is one-one and onto then it inverse will also be one-one and onto.
2. False- First if the function is surjective it's can't be possible. For an inverse function should be bijective.
A relation R⊆X×Y is left-total iff :
(∀s∈X) (∃t∈Y) [(s,t)∈R](∀s∈X) (∃t∈Y) [(s,t)∈R]
that is, iff every element of X relates to some element of Y.
So, here there can be a case that two elements of X relate to the same element of Y.
So it can't be always function.
Please let me know if I am incorrect.