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 the relationships \( f(g(x)) = |\sin x| \) and \( g(f(x)) = \sin^2 x \).

### Step-by-step Solution:

1. **Understanding the Given Functions:**
- We have two compositions: \( f(g(x)) = |\sin x| \) and \( g(f(x)) = \sin^2 x \).
- We need to express \( f(x) \) and \( g(x) \) in terms of \( x \).

2. **Let’s Assume \( g(x) \):**
- From \( g(f(x)) = \sin^2 x \), we can assume that \( g(x) \) is a function that transforms \( f(x) \) into \( \sin^2 x \).
- A reasonable assumption is that \( g(x) = \sin^2(x) \) when \( x \) is in the range of \( f \).

3. **Finding \( f(x) \):**
- Now, substituting \( g(x) \) into the first equation:
\[ f(g(x)) = f(\sin^2 x) = |\sin x| \]
- To satisfy this equation, we can assume that \( f(x) \) should be a function that, when applied to \( \sin^2 x \), gives \( |\sin x| \).
- A suitable function for \( f(x) \) is \( f(x) = \sqrt{x} \) because:
\[ f(\sin^2 x) = \sqrt{\sin^2 x} = |\sin x| \]

4. **Final Functions:**
- Therefore, we conclude:
\[ f(x) = \sqrt{x} \quad \text{and} \quad g(x) = \sin x \]

### Summary of Results:
- \( f(x) = \sqrt{x} \)
- \( g(x) = \sin x \)