If f∘g(x)= and g∘f(x)=∣sinx∣, then

(a)

Given Conditions:

        1.)    (f ∘ g) (x) = √x

⇒ f(g(x)) = √x (Equation 1)

        2.)    (g ∘ f) (x) = ∣sin x∣

                ⇒g(f(x)) = ∣sin x∣ (Equation 2)

Checking the Given Options:

Option (D): f(x)=x, g(x)=x

-> ​Compute f(g(x)) : f(g(x)) = f(√x) = √x

o   Satisfies f ∘ g(x)= x (Equation 1)

-> Compute g(f(x)): g(f(x)) = g(x) = √x

o   This must equal ∣sin x∣, but x ≠ ∣sin x∣ in general.

o   Does not satisfy Equation 2.

               Conclusion: Incorrect.

Option (B): f(x)=sin x

->Compute f(g(x)):  f(g(x)) = sin(g(x))

o   For this to be √x ​, we would need sin(g(x)) = x​, which isn't a general identity.

o   Does not satisfy Equation 1.

->Compute g(f(x)): g(f(x)) = g(sin x)

o   If g(x)=∣x∣, then g(f(x)) = ∣sin x∣, which satisfies Equation 2.

But since f(g(x)) = sin(g(x)) ≠ x, this option is incorrect.

Option (C): f(x) = x2, g(x)=sin x

->Compute f(g(x)):

o   f(g(x)) = f(sin x)=(sin x)2

o   This must equal √x​, but (sin x)2 does not match √x​ in general.

o   Does not satisfy Equation 1.

-> Compute g(f(x)): g(f(x)) = g(x2) = sin(x2)

o   This must equal ∣sin x∣, but sin(x2) ≠ ∣sin x∣ in general.

o   Does not satisfy Equation 2.

Conclusion: Incorrect.

Option (A): f and g cannot be determined.

->Since none of the options satisfy both conditions simultaneously, we cannot determine explicit functions f(x) and g(x) from the given choices.

Correct Answer: (A) f and g cannot be determined.