Domain of sqrt(f(x)): solve f(x) >= 0. Domain of log(f(x)): solve f(x) > 0. Domain of 1/f(x): solve f(x) != 0. For sqrt($ax^{2+bx+c}$) to be real for all x: need a>=0 and D<=0 (or a=0 with c>=0). These domain problems reduce to quadratic inequality solving.