If `fog = abs(sin x)` and gof =` sin^(2)x` then f(x) and g(x) are:

Options

A. `f(x)=sqrt(sinx),g(x)=x^(2)`
B. `f(x)=absx,g(x)=sinx`
C. `f(x)=sqrtx,g(x)=sin^(2)x`
D. `f(x)=sinsqrtx,g(x)=x^(2)`

Answer

To solve the problem, we need to find the functions \( f(x) \) and \( g(x) \) given that \( f \circ g = | \sin x | \) and \( g \circ f = \sin^2 x \).

### Step-by-step Solution:

1. **Understanding the Composition of Functions**:
We have two compositions:
- \( f(g(x)) = | \sin x | \)
- \( g(f(x)) = \sin^2 x \)

2. **Let’s Define \( g(x) \)**:
From the second equation \( g(f(x)) = \sin^2 x \), we can assume \( g(x) \) is a function that takes \( f(x) \) and outputs \( \sin^2 \) of that value.
Let's denote \( f(x) = t \), then we have:
\[ g(t) = \sin^2 t \]
Therefore, we can express \( g(x) \) as:
\[ g(x) = \sin^2 x \]

3. **Substituting \( g(x) \) into the First Equation**:
Now, we substitute \( g(x) = \sin^2 x \) into the first equation:
\[ f(g(x)) = f(\sin^2 x) = | \sin x | \]

4. **Finding \( f(x) \)**:
We need to find \( f(x) \) such that \( f(\sin^2 x) = | \sin x | \).
Let’s assume \( f(x) \) is a function of the form \( f(x) = \sqrt{x} \).
Then:
\[ f(\sin^2 x) = \sqrt{\sin^2 x} = | \sin x | \]
This satisfies the first equation.

5. **Final Functions**:
We have found:
\[ f(x) = \sqrt{x} \quad \text{and} \quad g(x) = \sin^2 x \]

### Conclusion:
Thus, the functions are:
\[ f(x) = \sqrt{x} \quad \text{and} \quad g(x) = \sin^2 x \]