Question

Consider the function $f\left(x\right)=e^x$ and $g\left(x\right)={\sin }^{-1}x$, then which of the following is/are necessarily true.

• A. Domain of $gof=$ Domain of $f$
• B. Range of $gof\subset$ Range of $g$
• C. Domain of $gof$ is $(-\infty ,0)$
• D. Range of $gof$ is $(-\frac{\pi }{2},0)$

Answer 1

The correct option is B Range of $gof\subset$ Range of $g$

$\text{solution:}f\left(x\right)=e^{x}g\left(x\right)={\sin }^{-1}\left(x\right)$

$\text{gof}={\sin }^{-1}\left(e^x\right)$

$\therefore -1⩽e^x⩽1$

$\text{so we need}x\in (-\infty \text{o]}$

$\text{domain of gof is}(-\infty ,0]$

$\text{Range of gof is same as range of}g$

$\text{Ansuer. : option (B)}$