Question

Arrange the following functions in the increasing order of their domain length. $f\left(x\right)=si{n}^{-1}x$,$g\left(x\right)=si{n}^{-1}\sqrt{2x}$, $h\left(x\right)={\sin }^{-1}2x$, $k\left(x\right)=ta{n}^{-1}\left(5x\right)$

• A. $f\left(x\right),g\left(x\right),h\left(x\right),k\left(x\right)$
• B. $g\left(x\right),h\left(x\right),k\left(x\right),f\left(x\right)$
• C. $g\left(x\right),h\left(x\right),f\left(x\right),k\left(x\right)$
• D. $g\left(x\right),f\left(x\right),h\left(x\right),k\left(x\right)$

Answer 1

Answer: (C)

$g\left(x\right),h\left(x\right),f\left(x\right),k\left(x\right)$

$f\left(x\right)={\sin }^{-1}x:x\in [-1,1]$
$g\left(x\right)={\sin }^{-1}\sqrt{2x}:\sqrt{2x}\in [-1,1]$
$\sqrt{2x}\in [-1,1]$
$2x\in [0,1]$
$x\in [0,\frac{1}{2}]$
$h\left(x\right)={\sin }^{-1}2x:2xϵ[-1,1]$
$x\in [\frac{-1}{2},\frac{1}{2}]$
$k\left(x\right)={\tan }^{-1}\left(5x\right):5x\in [-\infty ,\infty ]$
$x\in [-\infty ,\infty ]$

Increasing order of domain

$\Rightarrow g\left(x\right),h\left(x\right),f\left(x\right),k\left(x\right)$