If $f: R \rightarrow R^2$ and $g: R^{+} \rightarrow R$ are such that $g\{f(x)\}=|\sin x| \quad$ and $\quad…
If $f: R \rightarrow R^2$ and $g: R^{+} \rightarrow R$ are such that $g\{f(x)\}=|\sin x| \quad$ and $\quad f\{g(x)\}=(\sin \sqrt{x})^2$, then a possible choice for $f$ and $g$ is
$f(x)=x^2, g(x)=\sin \sqrt{x}$
$f(x)=\sin x, g(x)=|x|$
$f(x)=\sin ^2 x, g(x)=\sqrt{x}$
$f(x)=x^2, g(x)=\sqrt{x}$
Solution
Given, $g\{f(x)\}=|\sin x|$
and $\quad f\{g(x)\}=(\sin \sqrt{x})^2$
Let us consider $f(x)=\sin ^2 x$ and $g(x)=\sqrt{x}$ $\therefore \quad f\{g(x)\}=f(\sqrt{x})=\left(\sin ^2 \sqrt{x}\right)=(\sin \sqrt{x})^2$ and $g\{f(x)\}=g\left(\sin ^2 x\right)=\sqrt{\sin ^2 x}=|\sin x|$