The range of the function $f\left(x\right)={\sin }^{-1}\left[x^2+\frac{1}{2}\right]+{\cos }^{-1}\left[x^2-\frac{1}{2}\right]$ where $\left[\right]$  denotes greatest integer function

$\left[x^2-\frac{1}{2}\right]=\left[x^2+\frac{1}{2}-1\right]=\left[x^2+\frac{1}{2}\right]-1$

$f\left(x\right)={\sin }^{-1}\left[x^2+\frac{1}{2}\right]+{\cos }^{-1}\left(\left(x^2+\frac{1}{2}\right)-1\right)$

$\left[x^2+\frac{1}{2}\right]=0,1$

$f\left(x\right)={\sin }^{-1}\left(0\right)+{\cos }^{-1}\left(-1\right)$ or ${\sin }^{-1}\left(1\right)+{\cos }^{-1}\left(0\right)$

$\therefore f\left(x\right)=\pi$