## Domain of a Function

$\displaystyle \begin{array}{l}1.\,\,\,f(x)=\frac{p(x)}{q(x)}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Rule:\,q(x)\ne 0\\2.\,\,\,f(x)=\sqrt{p(x)}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Rule:p(x)\ge 0\\3.\,\,\,f(x)=\frac{p(x)}{\sqrt{q(x)}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Rule:q(x)>0\\4.\,\,\,f(x)=\left| x \right|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Rule:x\in R\\5.\,\,\,f(x)={{\log }_{p(x)}}q(x)\,\,\,\,\,\,\,\,\,\,\,Rule:p(x)>0\,\,and\,\,q(x)>0\,\,and\,\,p(x)\ne 1\end{array}$