If g (f (x)) = lsin xl and f (g(x)) =` (sin sqrt(x))^2`, then

Options

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 and g cannot be determined

Answer

To solve the problem, we need to find the functions \( f(x) \) and \( g(x) \) given the equations:

1. \( g(f(x)) = |\sin x| \)
2. \( f(g(x)) = (\sin(\sqrt{x}))^2 \)

We will analyze the options to identify the correct functions.

### Step 1: Analyze the first equation \( g(f(x)) = |\sin x| \)

Let's assume \( f(x) = \sin^2 x \). Now we need to find \( g(x) \) such that:

\[ g(f(x)) = g(\sin^2 x) = |\sin x| \]

If we let \( g(x) = \sqrt{x} \), then:

\[ g(f(x)) = g(\sin^2 x) = \sqrt{\sin^2 x} = |\sin x| \]

This satisfies the first equation.

### Step 2: Analyze the second equation \( f(g(x)) = (\sin(\sqrt{x}))^2 \)

Now we need to check if \( f(g(x)) \) gives us \( (\sin(\sqrt{x}))^2 \) when \( g(x) = \sqrt{x} \).

Substituting \( g(x) = \sqrt{x} \) into \( f(x) \):

\[ f(g(x)) = f(\sqrt{x}) = \sin^2(\sqrt{x}) \]

This matches the second equation:

\[ f(g(x)) = \sin^2(\sqrt{x}) = (\sin(\sqrt{x}))^2 \]

### Conclusion

Both equations are satisfied with the following functions:

- \( f(x) = \sin^2 x \)
- \( g(x) = \sqrt{x} \)

Thus, the final answer is:

\[ f(x) = \sin^2 x, \quad g(x) = \sqrt{x} \]