# If g (f (x))=|sin x| and f (g (x))

Question:

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

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

(b) $f(x)=\sin x, g(x)=|x|$

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

(d) $f$ and $g$ cannot be determined.

Solution:

If we solve it  by the trial-and-error method, we can see that (a) satisfies the given condition.
From (a):

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

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