Question

If $f\left(x\right)={\sin }^{2}x$ and the composite function $g\left(f\right(x\left)\right)=\left|\sin x\right|$, then $g\left(x\right)$ is equal to

• A. $\sqrt{x-1}$
• B. $\sqrt{x}$
• C. $\sqrt{x+1}$
• D. $-\sqrt{x}$

Answer 1

The correct option is B $\sqrt{x}$

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

$A:$ Take $g\left(x\right)=\sqrt{x-1}$

$\Rightarrow$ $g\left(f\right(x\left)\right)=g\left[f\right(x\left)\right]$

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

$=\sqrt{{\sin }^{2}x-1}$

We can see, $g\left(x\right)=\sqrt{x-1}$ is not possible.

$B:$ Take $g\left(x\right)=\sqrt{x}$

$\Rightarrow$ $g\left(f\right(x\left)\right)=g\left[f\right(x\left)\right]$

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

$=\sqrt{{\sin }^{2}x}$

$=\pm \sin x$

$=|\sin x|$

$\therefore$ Here, we can see $g\left(x\right)=\sqrt{x}$.