to find range put f(x)=y then make a quadratic equation w.r.t x, after that apply condition for real roots and solve the inequality to get value of range.
the critical points of range act as extremum for f(x) so from there you get extremum. and check at any value of x does denominator of f(x) becomes 0. there it will be not differentiable.
find the derivative of f(x) and check wheather it is only positive or only negative for all x of it is both. if only positive/negative it will be one one function otherwise it is many one.
for onto check Range=codomain
