[tex]\displaystyle \sf f(x)=\sqrt{\log_5(x^2-1) } \ \ ;\ D(f) =\ ? \\\\\\ \text{Raiz quadrado nos reais \'e maior ou igual a zero, ent\~ao}: \\\\ \log_5(x^2-1)\geq 0\\\\ \text{Condi\c c\~ao de exist\^encia de um log: logaritmando maior que zero} \\\\ x^2-1 > 0 \to x^2 > 1\to \boxed{\sf \ x > 1\ ou\ x < -1 } \\\\ \text{temos}: \\\\ \log_5(x^2-1)\geq 0 \\\\ x^2-1\geq 5^0 \\\\ x^2-1\geq 1 \\\\ x^2\geq 1+ 1\to x^2 \geq 2 \\\\ \boxed{\sf x\geq \sqrt{2}\ \ ou\ \ x\leq -\sqrt{2}}[/tex]

[tex]\displaystyle \sf \text{Fazendo interse\c c\~ao com a condi\c c\~ao de exist\^encia} :\\\\ D(f) = (x\geq \sqrt{2}\ ou\ x\leq-\sqrt{2})\cap (x > 1\ ou \ x < -1) \\\\\\ \boxed{\begin{matrix}\\ \ \displaystyle \sf D(f) = \left\{x\in\mathbb{R}\ |\ x\leq -\sqrt{2}\ ou\ x\geq \sqrt{2} \ \right\}\\\\ \sf OU \\\\ \displaystyle \sf D(f) = \left (-\infty,-\sqrt{2}\ \right] \cup \left \left[\sqrt{2},\ +\infty\right)\\ _ \end{matrix}\ }\Large{\checkmark }[/tex]