Question

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

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

Answer 1

The correct option is A

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

Let $f\left(x\right)=si{n}^{2}x$ and $g\left(x\right)\sqrt{x}$
Now, $fog\left(x\right)=f\left[g\right(x\left)\right]=f\left(\sqrt{x}\right)=si{n}^2\sqrt{x}$
and $gof\left(x\right)=g\left[f\right(x\left)\right]=g\left(si{n}^{2}x\right)=\sqrt{si{n}^{2}x}=|sinx|$
Again, let $f\left(x\right)=sinx,g\left(x\right)=|x|$
$fog\left(x\right)=f\left[g\right(x\left)\right]=f\left(|x|\right)=sin|x|\ne (sin\sqrt{x}{)}^2$
When, $f\left(x\right)=x^2,g\left(x\right)=sin\sqrt{x}$
$fog\left(x\right)=f\left[g\right(x\left)\right]=f\left(sin\sqrt{x}\right)=(sin\sqrt{x}{)}^2$
and $\left(gof\right)\left(x\right)=g\left[f\right(x\left)\right]=g\left(x^2\right)=sin\sqrt{x^2}$
$=sin|x|\ne |sinx|$