Question

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

• A. $\left[-\frac{1}{3},1\right]$
• B. $\left[\frac{1}{2},1\right]$
• C. $\left[-\frac{1}{2},\frac{1}{3}\right]$
• D. $\left[-1,\frac{1}{3}\right]$
• E. $\left[-\frac{1}{3},\frac{1}{2}\right]$

Answer 1

Answer: (E)

The given function is $f(x)=\sqrt{1-2x}+2{\sin }^{-1}\left(\frac{3x-1}{2}\right)$
For domain of $\sqrt{1-2x,}1-2x\ge 0$
$\Rightarrow$ $1\ge 2x\Rightarrow x\le \frac{1}{2}$
$\Rightarrow$ $x\in \left(-\infty ,\frac{1}{2}\right]$ and for domain of $2{\sin }^{-1}\left(\frac{3x-1}{2}\right),$ $-1\le \frac{3x-1}{2}\le 1$
$\Rightarrow$ $-2\le 3x-1\le 2$
$\Rightarrow$ $-2+1\le 3x\le 2+1$
$\Rightarrow$ $-1\le 3x\le 3$
$\Rightarrow$ $-\frac{1}{3}\le x\le 1$
$\therefore$ Domain of $f(x)=\left[-\frac{1}{3},\frac{1}{2}\right]$