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Question
If \[g \left( f \left( x \right) \right) = \left| \sin x \right| \text{and} f \left( g \left( x \right) \right) = \left( \sin \sqrt{x} \right)^2 , \text{then}\]
Options
\[f\left( x \right) = \sin^2 x, g\left( x \right) = \sqrt{x}\]
\[f\left( x \right) = \sin x, g\left( x \right) = |x|\]
\[f\left( x \right) = x^2 , g\left( x \right) = \sin \sqrt{x}\]
f and g cannot be determied
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Solution
If we solve it by the trial-and-error method, we can see that (a) satisfies the given condition.
From (a):
\[f\left( x \right) = \sin^2 x \text{ and } g\left( x \right) = \sqrt{x}\]
\[ \Rightarrow f\left( g\left( x \right) \right) = f\left( \sqrt{x} \right) = \sin^2 \sqrt{x} = \left( \sin \sqrt{x} \right)^2\]
So, the answer is (a).
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