Question

The number of integral point(s) in the domain of $f\left(x\right)={\sin }^{-1}\left[\frac{x}{5}\right]+{\cos }^{-1}\left\{x\right\}$ is
(where $[.]$ denotes greatest integer function and $\{.\}$ denotes fractional part function)

Answer 1

Given : $f\left(x\right)={\sin }^{-1}\left[\frac{x}{5}\right]+{\cos }^{-1}\left\{x\right\}$
We know that, $\left\{x\right\}\in [0,1)$
So, ${\cos }^{-1}\left\{x\right\}$ is defined for all $x\in R$

Now, check domain of ${\sin }^{-1}\left[\frac{x}{5}\right]:$
$-1\le \left[\frac{x}{5}\right]\le 1$
$\Rightarrow -1\le \frac{x}{5}<2$
$\Rightarrow -5\le x<10$
$\Rightarrow$ Domain of $f\left(x\right)$ is $[-5,10).$
So, number of integral points in domain is $15.$