Question

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

• A. $f\left(x\right)={\sin }^{2}x,g\left(x\right)=\sqrt{x}$
• B. $f\left(x\right)=\sin x,g\left(x\right)=|x|$
• C. $f\left(x\right)=x^2,g\left(x\right)=\sin \sqrt{x}$
• D. $f,g$ cannot be determined

Answer 1

Answer: (A)

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

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

Comparing $i$ and $ii$

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

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

And

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

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

Therefore

$\Rightarrow$$g\left(x\right)=\sqrt{x}$

$\Rightarrow$$f\left(x\right)={\sin }^{2}x$