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 $g(x)=\sqrt{x}$
$\Rightarrow f(g(x))=f(\sqrt{x})=\sin ^{2} \sqrt{x}=(\sin \sqrt{x})^{2}$
So, the answer is (a).
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