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:

When solving composite function equations, substitute each option into both given equations to check consistency. Ensure the domain and range align with the problem constraints.

Answer 1

The Correct Option is A

Step 1: Analyze the given functional equations.
We are given two composite functions:
\( g(f(x)) = |\sin x| \),
\( f(g(x)) = (\sin \sqrt{x})^2 \).
We need to find functions \( f(x) \) and \( g(x) \) that satisfy both equations simultaneously.

Step 2: Substitute and test option (A).
Consider option (A): \( f(x) = \sin^2 x \), \( g(x) = \sqrt{x} \).
Compute \( g(f(x)) \): \[ g(f(x)) = g(\sin^2 x) = \sqrt{\sin^2 x}. \] Since \( \sqrt{\sin^2 x} = |\sin x| \) (as the square root of a square is the absolute value), we get: \[ g(f(x)) = |\sin x|, \] which matches the first equation. Compute \( f(g(x)) \): \[ f(g(x)) = f(\sqrt{x}) = \sin^2 \sqrt{x}. \] We need to check if \( \sin^2 \sqrt{x} = (\sin \sqrt{x})^2 \). Since \( (\sin \sqrt{x})^2 \) is the square of \( \sin \sqrt{x} \), and \( \sin^2 \sqrt{x} \) is the same (as \( (\sin \theta)^2 = \sin^2 \theta \)), we have: \[ f(g(x)) = \sin^2 \sqrt{x} = (\sin \sqrt{x})^2, \] which matches the second equation.
Step 3: Verify consistency.
The domain of \( g(x) = \sqrt{x} \) requires \( x \geq 0 \), and since \( |\sin x| \) and \( (\sin \sqrt{x})^2 \) are defined for all real \( x \geq 0 \), the functions are consistent.
Option (A) satisfies both equations, confirming it as a valid solution.

Step 4: Test other options to ensure uniqueness.
% Option (B) \( f(x) = \sin x \), \( g(x) = |x| \):
\( g(f(x)) = g(\sin x) = |\sin x| \),
\( f(g(x)) = f(|x|) = \sin |x| \), not \( (\sin \sqrt{x})^2 \).
% Option (C) \( f(x) = x^2 \), \( g(x) = \sin \sqrt{x} \):
\( g(f(x)) = g(x^2) = \sin \sqrt{x^2} = \sin |x| \), not \( |\sin x| \) unless \( x \geq 0 \),
\( f(g(x)) = f(\sin \sqrt{x}) = (\sin \sqrt{x})^2 \), which matches, but \( g(f(x)) \) fails.
% Option (D) \( f(x) = |x| \), \( g(x) = \sin x \):
\( g(f(x)) = g(|x|) = \sin |x| = |\sin x| \),
\( f(g(x)) = f(\sin x) = |\sin x| \), not \( (\sin \sqrt{x})^2 \).

Step 5: Conclude the result.
Option (A) satisfies both \( g(f(x)) = |\sin x| \) and \( f(g(x)) = (\sin \sqrt{x})^2 \), making it the correct solution.