Question

Find the domain of definition of the following function:
$y=\text{arc}\sin \frac{x-3}{2}-\log (4-x).$

Answer 1

We know domain of ${\sin }^{-1}$ is $[-1,1]$

For

${\sin }^{-1}\frac{x-3}{2}-1\le \frac{x-3}{2}\le 1-2\le x-3\le 21\le x\le 5$

and

$\log (4-x)4-x>04>x$

Taking intersection, we get

$x\in [1,4)$