First part, answered here.
As for the range, since hence, we require that thus the range is (The domain is all reals, of course.)
Strictly speaking does NOT have an inverse because it is not a "one to one" function.
Yes, you can recognize that the graph of y= f(x) has a vertex at (1, 2) and then break it into TWO functions:
with domain which then has range and write or
with domain which also has range and write .
But neither of those functions is f(x) so, strictly speaking, neither of those is the inverse of f.