Largest real domains

The domain is the set of permitted inputs. For the largest real domain, exclude precisely the inputs at which the original rule is not defined. A square root needs a non-negative radicand; a logarithm needs a positive argument; every denominator must be non-zero. A square root in a denominator therefore needs a strictly positive radicand. Write each condition from the original formula, solve it, then keep only inputs satisfying every condition. Do this before cancelling factors. For h(x)= sqrt(x+2)x-1, require x+2 0 and x-1 0. Thus D_ h=[-2,1) (1,infinity). Check: x=-2 is allowed because its numerator is zero; x=1 is forbidden. Cancelling a factor does not restore an input missing from the original rule.