Question

Find the domain of definition of the following function:
$y=\sqrt{\log \frac{5x-x^2}{4}}$

Answer 1

$y=\sqrt{log\frac{(5x-x)}{4}}$

For $y$ to be real and defined

Therefore,$\frac{5x-x^2}{4}>0\Rightarrow x^2-5x<0$

Therefore, $x\in (0,5)⟶\left(1\right)$

Therefore, $log\left(\frac{5x-x^2}{4}\right)\ge 0\Rightarrow \frac{5x-x^2}{4}\ge 1\Rightarrow x^2-5x+4\le 0$

$x\in (1,4)⟶\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$, $x\in [1,4]$

Therefore,domain of $y$ is $x\in [1,4]$