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

$\{π\}$

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

$-1≤\left(x^2+\frac{1}{2}\right)<2$,    $-1≤\left(x^2-\frac{1}{2}\right)<2$

$\frac{-3}{2}≤x^2<\frac{3}{2}$,    $\frac{-1}{2}≤x^2<\frac{5}{2}$

$⇒0≤x^2<\frac{3}{2}$,    $⇒0≤x^2<\frac{5}{2}$

taking intersection

$x^2∈[0,\frac{3}{2})$

as $x^2∈[0,\frac{1}{2})$

$f(x)=\sin^{-1}(0)+\cos^{-1}(-1)=π$

as $x^2∈\left[\frac{1}{2},\frac{3}{2}\right)$

$f(x)=\cos^{-1}(0)+\sin^{-1}(1)=π$

so $f(x)∈\{π\}$