Question

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

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

Answer 1

The correct option is A

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

Explanation for the correct option:

Find the value of $f\left(x\right)$ and $g\left(x\right)$:

Given that,

$g\left(f\left(x\right)\right)=\left|\sin x\right|$

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

Let's check from option A.

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

Now, find $f\left(g\left(x\right)\right)$ and $g\left(f\left(x\right)\right)$,

$\begin{array}{rcl}f\left(g\left(x\right)\right)& =& f\left(\sqrt{x}\right)\\ & =& {\sin }^2\sqrt{x}\end{array}$ and $\begin{array}{rcl}g\left(f\left(x\right)\right)& =& g\left({\sin }^{2}x\right)\\ & =& \sqrt{{\sin }^{2}x}\\ & =& \left|\sin x\right|\left[\because \sqrt{a}=\left|a\right|\right]\end{array}$

Since, option A satisfies the given condition.

Hence, option A is correct.

Explanation for the incorrect option:

For option B.

$f\left(x\right)=\sin x$ and $g\left(x\right)=\left|x\right|$

Now, find $f\left(g\left(x\right)\right)$ and $g\left(f\left(x\right)\right)$,

$\begin{array}{rcl}f\left(g\left(x\right)\right)& =& f\left(\left|x\right|\right)\\ & =& \sin \left|x\right|\end{array}$ and $\begin{array}{rcl}g\left(f\left(x\right)\right)& =& g\left(\sin x\right)\\ & =& \left|\sin x\right|\end{array}$

Since, option B does not satisfy the given condition.

Hence, option B is not correct.

For option C.

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

Now, find $f\left(g\left(x\right)\right)$ and $g\left(f\left(x\right)\right)$,

$\begin{array}{rcl}f\left(g\left(x\right)\right)& =& f\left(\sin \sqrt{x}\right)\\ & =& {\left(\sin \sqrt{x}\right)}^2\end{array}$ and $\begin{array}{rcl}g\left(f\left(x\right)\right)& =& g\left(x^2\right)\\ & =& \sin \sqrt{x^2}\\ & =& \sin \left|x\right|\end{array}$

Since, option C also does not satisfy the given condition.

Hence, option C is not correct.

Hence, the correct option is A.