Question

$f\left(x\right)=\sqrt{\log \left(\frac{3x-x^2}{x-1}\right)}$

Answer 1

$f\left(x\right)=\sqrt{log\left(\frac{3x-x^2}{x-1}\right)}$

$\Rightarrow log\frac{3x-x^2}{x-1}\ge 0$

$\frac{3x-x^2}{x-1}\ge 1$

$\Rightarrow \frac{3x-x^2-x+1}{x-1}\ge 0$ $\Rightarrow \frac{-x^2+2x+1}{x-1}\ge 0$ $\frac{x^2-2x-1}{(x-1)}\le 0$

$\frac{3x-x^2}{x-1}\ge 0$

$\frac{x^2-3x}{x-1}<0$ $\Rightarrow \frac{x(x-3)}{(x-1)}<0$.

$x\in (\infty ,0)\cup (1,3)$ $x\in (\infty ,0)$.