Problem :
Find the range of $f(x) = \sin^{-1}x +\tan^{-1}x +\cos^{-1}x$
Solution : Since, $\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$
Since range of $\tan^{-1}x$ is $ (\frac{-\pi}{2}, \frac{\pi}{2})$
$\therefore, \frac{-\pi}{2} \leq \tan^{-1}x \leq \frac{\pi}{2}$
= $ \frac{-\pi}{2} + \frac{\pi}{2} \leq \tan^{-1}x + \frac{\pi}{2} \leq \frac{\pi}{2} + \frac{\pi}{2}$.
= $0 \leq \tan^{-1}x+ \frac{\pi}{2} \leq \pi $
Is it correct.. please suggest thanks….
Best Answer
Given $f(x) = \sin^{-1}(x)+\cos^{-1}(x)+\tan^{-1}(x)$
First we will calculate domain of function $f(x)$
function $\sin^{-1}(x)$ is defined in $\displaystyle x\in \left[-1,1\right]$ Similarly function $\cos^{-1}(x)$ is defined in $\displaystyle x\in \left[-1,1\right]$
and function $\tan^{-1}(x)$ is defined in $\displaystyle x\in \left(-\infty,+\infty\right)$
So $f(x) = \sin^{-1}(x)+\cos^{-1}(x)+\tan^{-1}(x)$ is defined in $\displaystyle x\in \left[-1,1\right]$
So $f(x) = \sin^{-1}(x)+\cos^{-1}(x)+\tan^{-1}(x)$ is defined in $$\displaystyle x\in \left[-1,1\right]$$
So $\displaystyle f(x) = \frac{\pi}{2}+\tan^{-1}(x)$
Now $\displaystyle f^{'}(x) = \frac{1}{1+x^2}>0\;\forall x\in [-1,1]$
So $f(x)$ is Strictly Increasing function.
So $\displaystyle f(-1) = \frac{\pi}{2}+\tan^{-1}(-1) = \frac{\pi}{2}-\frac{\pi}{4} = \frac{\pi}{4}$
and $\displaystyle f(+1) = \frac{\pi}{2}+\tan^{-1}(1) = \frac{\pi}{2}+\frac{\pi}{4} = \frac{3\pi}{4}$
So $\displaystyle f(x)\in \left[\frac{\pi}{4}\;,\frac{3\pi}{4}\right]$