Question

Find the domain of the following function:

$f\left(x\right)={\sin }^{-1}\left(\frac{2-3\left[x\right]}{4}\right),$ which $[\cdot ]$ denotes the greatest integer function, the domain is $[0,k),k$ is ?

Answer 1

We know, for ${\sin }^{-1}x$ to be defined $-1\le x\le 1$
Thus, for given function to be defined,
$-1\le \frac{2-3\left[x\right]}{4}\le 1$
$\Rightarrow -4\le 2-3\left[x\right]\le 4$
$\Rightarrow -6\le -3\left[x\right]\le 2$
$\Rightarrow -2/3\le \left[x\right]\le 2$
But we know $\left[x\right]$ is always an integer
$\Rightarrow 0\le \left[x\right]\le 3$

$\therefore x\in [0,3)$