Question

The domain of the function defined by$f\left(x\right)={\sin }^{-1}\sqrt{x-1}$ is

• A. $[-1,1]$
• B. None of the these
• C. $[1,2]$
• D. $[0,1]$

Answer 1

The correct option is C $[1,2]$

Given:$f\left(x\right)={\sin }^{-1}\sqrt{x-1}$

To find : Domain of $f\left(x\right)$

$\because$ Domain of ${\sin }^{-1}x\text{is}[-1,1]$

$\Rightarrow -1\le \sqrt{x-1}\le 1$

$\Rightarrow 0\le \sqrt{x-1}\le 1$ $[\because \sqrt{x}\ge 0]$

$\Rightarrow 0\le x-1\le 1$

$\Rightarrow 1\le x\le 2$ $\dots \dots \left(i\right)$

Also, $x-1\ge 0$

$[\because \sqrt{x-1}\text{should also be defined}]$

$\Rightarrow x\ge 1$ $\dots ..\left(ii\right)$

Taking intersection of $\left(i\right)\&\left(ii\right)$

$1\le x\le 2$

$\Rightarrow \text{The domain of}{\sin }^{-1}\sqrt{x-1}\text{is}[1,2]$

Therefore, option $\left(a\right)$ is correct.