Question:
The domain of the function
$f(x)=\sin ^{-1}\left(\frac{3 x^{2}+x-1}{(x-1)^{2}}\right)+\cos ^{-1}\left(\frac{x-1}{x+1}\right)$ is :
Correct Option: , 3
Solution:
$f(x)=\sin ^{-1}\left(\frac{3 x^{2}+x-1}{(x-1)^{2}}\right)+\cos ^{-1}\left(\frac{x-1}{x+1}\right)$
$-1 \leq \frac{x-1}{x+1} \leq 1 \Rightarrow 0 \leq x<\infty$........(1)
$-1 \leq \frac{3 x^{2}+x-1}{(x-1)^{2}} \leq 1 \Rightarrow x \in\left[\frac{-1}{4}, \frac{1}{2}\right] \cup\{0\}$......(2)
(1) & (2)
$\Rightarrow$ Domain $=\left[\frac{1}{4}, \frac{1}{2}\right] \cup\{0\}$