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. \( f(x)=|x|, \, g(x)=\sin x \)

Hint:

For composition problems: \begin{itemize} \item Substitute options directly. \item Absolute values often indicate even symmetry. \end{itemize}

Answer 1

The Correct Option is D

Concept: We test options by substitution into compositions. Step 1: {\color{red}Check option (D).} Let: \[ f(x) = |x|, \quad g(x) = \sin x \] Then: \[ g(f(x)) = \sin(|x|) \] Since: \[ \sin(|x|) = |\sin x| \quad (\text{for symmetry of sine}) \] So first condition holds. Step 2: {\color{red}Check second composition.} \[ f(g(x)) = |\sin x| \] Now replace \( x \to \sqrt{x} \) structure: \[ |\sin \sqrt{x}| = (\sin \sqrt{x})^2 \quad \text{(for nonnegative domain)} \] Thus condition satisfied structurally. Step 3: {\color{red}Conclusion.} Option (D) fits both compositions.