Question

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

• A. $f\left(x\right)$ and $g\left(x\right)$ are both periodic functions.
• B. $f\left(x\right)$ is periodic but $g\left(x\right)$ is not a periodic function
• C. $f\left(x\right)$ is not periodic while $g\left(x\right)$ is a periodic function
• D. $f\left(x\right)$ and $g\left(x\right)$ are both non-periodic functions.

Answer 1

The correct option is C $f\left(x\right)$ is not periodic while $g\left(x\right)$ is a periodic function

We know that $\sqrt{a^2}=|a|$
$\Rightarrow f\left(g\right(x\left)\right)=|\sin x|=\sqrt{{\sin }^{2}x}$
$\Rightarrow f\left(x\right)=\sqrt{x};g\left(x\right)={\sin }^{2}x$
$\Rightarrow f\left(x\right)$ is non periodic and $g\left(x\right)$ is periodic.