Question

If $fog=|\sin x|$ and $gof={\sin }^2\sqrt{x}$, then $f\left(x\right)$ and $g\left(x\right)$ are

• A. $f\left(x\right)=\sqrt{\sin x},g\left(x\right)=x^2$
• B. $f\left(x\right)=|x|,g\left(x\right)=\sin x$
• C. $f\left(x\right)=\sqrt{x},g\left(x\right)={\sin }^{2}x$
• D. $f\left(x\right)=\sin \sqrt{x},g\left(x\right)=x^2$

Answer 1

Answer: (C)

$f\left(x\right)=\sqrt{x},g\left(x\right)={\sin }^{2}x$

$fog=f\left\{g\right(x\left)\right\}=|\sin x|=\sqrt{{\sin }^{2}x}$

Also, $gof=g\left\{f\right(x\left)\right\}={\sin }^2\sqrt{x}$

Obviously, $\sqrt{{\sin }^{2}x}=\sqrt{g\left(x\right)}$

and ${\sin }^2\sqrt{x}={\sin }^2\left\{f\right(x\left)\right\}$
i.e., $g\left(x\right)={\sin }^{2}x$

and $f\left(x\right)=\sqrt{x}$.