Domain determination often reduces to quadratic inequalities: sqrt($ax^{2+bx+c}$) requires $ax^{2+bx+c}$ >= 0. $log_b$(f(x)) requires f(x)>0 and b>0, b!=1. 1/sqrt(f(x)) requires f(x)>0 (strictly positive). For sqrt(|x|-x): if x>=0, |x|-x=0 (valid, gives 0); if x<0, |x|-x=-x-x=-2x>0 (valid). So domain = all real numbers. Combining multiple conditions uses intersection. These are application problems where modulus and quadratic inequality skills directly apply.