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 :

• A. $\left[0, \frac{1}{4}\right]$
• B. $[-2,0] \cup\left[\frac{1}{4}, \frac{1}{2}\right]$
• C. $\left[\frac{1}{4}, \frac{1}{2}\right] \cup\{0\}$
• D. $\left[0, \frac{1}{2}\right]$

Answer 1

Answer: (C)

$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 $=\left[\frac{1}{4}, \frac{1}{2}\right] \cup\{0\}$