Question

The domain of the function $y=\frac{1}{\sqrt{{81}^{1/(x-1)}-3}}$ is

• A. $(3,4)$
• B. $(3,5)$
• C. $(1,5)$
• D. $(3,6)$

Answer 1

The correct option is C $(1,5)$

For $f\left(x\right)$ to be defined ${81}^{\left(\frac{1}{x-1}\right)}-3>0$
$\Rightarrow 3^{\left(\frac{4}{x-1}\right)}>3^1$
$\Rightarrow \frac{4}{x-1}>1$
$\Rightarrow \frac{4}{x-1}-1>0$
$\Rightarrow \frac{5-x}{x-1}>0$
$\Rightarrow \frac{x-5}{x-1}<0$
$\Rightarrow x\in (1,5)$