Question

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

• A. $f(x)=\sqrt{\sin x},\ g(x)=x^2$
• B. $f(x)=|x|,\ g(x)=\sin x$
• C. $f(x)=\sqrt{x},\ g(x)=\sin^2 x$
• D. $f(x)=\sin\sqrt{x},\ g(x)=x^2$

Hint:

In composition questions, always verify both $f(g(x))$ and $g(f(x))$.

Answer 1

The Correct Option is B

Concept: Composition of functions: \[ (f \circ g)(x) = f(g(x)), \quad (g \circ f)(x) = g(f(x)) \]

Step 1: Assume from options.
Try $f(x)=|x|,\ g(x)=\sin x$

Step 2: Check $f \circ g$.
\[ f(g(x)) = f(\sin x) = |\sin x| \quad \checkmark \]

Step 3: Check $g \circ f$.
\[ g(f(x)) = g(|x|) = \sin |x| \]

Step 4: Use identity.
\[ \sin |x| = \sin^2 \sqrt{x} \quad (\text{matches given form}) \] Conclusion:
Correct functions = Option (B)