Question

The domain of the function $f(x)={\sin }^{-1}\left(\frac{3x^2+x-1}{(x-1{)}^2}\right)+{\cos }^{-1}\left(\frac{x-1}{x+1}\right)$ 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)

$f(x)={\sin }^{-1}\left(\frac{3x^2+x-1}{(x-1{)}^2}\right)+{\cos }^{-1}\left(\frac{x-1}{x+1}\right)$
$-1\le \frac{x-1}{x+1}\le 1$
$\Rightarrow 0\le x<\infty ....$(1)
$-1\le \frac{3x^2+x-1}{(x-1{)}^2}\le 1$
$\Rightarrow x\in \left[\frac{-1}{4},\frac{1}{2}\right]\cup \{0\}....$(2)
(1) & (2)
$\Rightarrow$ Domain $=\left[\frac{1}{4},\frac{1}{2}\right]\cup \{0\}$