Question

Find the domains of definition of the following function:
$f\left(x\right)=\frac{\sqrt{3{\log }_{64}x-1}}{\sqrt[3]{32x-11}}.$

Answer 1

$f\left(x\right)=\frac{\sqrt{3lo{g}_{64}x-1}}{\sqrt[3]{32x-11}}$

For $y$ to be real and finite,

$x$ must be $>0$

$\Rightarrow x>0$

${3log}_{64}x-1\ge 0\Rightarrow {log}_{64}x\ge \frac{1}{3}\Rightarrow x\ge 4$

$2x-11\ne 0\Rightarrow x\ne \frac{11}{2}$

Therefore, domain of $f\left(x\right)$ is $x\in [4,\frac{11}{2})\cup (\frac{11}{2},\infty ).$