Question

Which pair of functions is identical?

• A. ${\sin }^{-1}\left(\sin x\right)$ and $\sin \left({\sin }^{-1}x\right)$
• B. ${\log }_e{e}^x,{\log }^{lo{g}_{e}x}$
• C. ${\log }_e{x}^2,2{\log }_{e}x$
• D. none of these

Answer 1

Answer: (D)

none of these

Here, $\left(a\right){\sin }^{-1}\left(\sin x\right)$ is defined for $x\in [-\frac{\pi }{2},\frac{\pi }{2}]$, while $\sin \left({\sin }^{-1}x\right)$ is defined only for $x\in [-1,1]$
(b) ${\log }_e{e}^x$ is defined for all $x,$ while $e^{{\log }_{e}x}$ is defined for $x>0.$
(c) ${\log }_e{x}^2$ is defined for all $xϵR-\left\{0\right\}$, while $2\log x$ is defined for $x>0$
$\therefore$ None of identical.
Hence, (d) is the correct answer.