The range of the function 

$\mathrm{f}\left(\mathrm{x}\right)={\sin }^{-1}\mathrm{x}+{\cos }^{-1}\mathrm{x}+{\tan }^{-1}\mathrm{x}$ is

$$\begin{gathered} \mathrm{The} \mathrm{domain} \mathrm{of} \mathrm{f}\left(\mathrm{x}\right)=\left[-1,1\right] \\ We know that {\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2} \\ Given function becomes f\left(x\right)=\frac{\pi }{2}+{\tan }^{-1}x \\ Range of f\left(x\right)=\left[\frac{\pi }{2}-\frac{\mathrm{\pi }}{4},\frac{\pi }{2}+\frac{\mathrm{\pi }}{4}\right] \\ \mathrm{The} \mathrm{range} \mathrm{of} \mathrm{f}\left(\mathrm{x}\right)=\left[\frac{\mathrm{\pi }}{4}, \frac{3\mathrm{\pi }}{4}\right] \end{gathered}$$