The range of f (x) = 4 ^ - 1 ( x ^ 2 x ^ 2 + 1 ) is

Given,

$f\left(x\right) = 4\sin^{- 1}\left(\dfrac{{x}^{2}}{{x}^{2} + 1}\right)$

Now taking, $\dfrac{{x}^{2}}{1 + {x}^{2}} = 1 - \dfrac{1}{1 + {x}^{2}} < 1$

$⇒ 0 \le \dfrac{{x}^{2}}{1 + {x}^{2}} < 1$

Now taking $\sin^{- 1}$ we get,

$⇒ 0 \le \sin^{- 1}\left(\dfrac{{x}^{2}}{1 + {x}^{2}}\right) < \dfrac{\pi}{2}$

$⇒ 0 \le 4\sin^{- 1}\left(\dfrac{{x}^{2}}{1 + {x}^{2}}\right) < 2\pi$

Hence, the range of $4\sin^{- 1}\left(\dfrac{{x}^{2}}{{x}^{2} + 1}\right)$ is $[ 0, 2\pi )$