If `f(x)={{:(-1, x lt 0),(0, x=0 and g(x)=x(1-x^(2))", then"),(1, x gt 0):}`

Options

A. `fog(x)={{:(-1, -1 lt x lt 0 or x gt 1),(0, x =0 , 1,-1),(1, 0 lt x lt 1):}`
B. `fog(x)={{:(-1, -1 lt x lt 0),(0, x=0,1,-1),(1, 0 lt x lt 1):}`
C. `fog(x)={{:(-1, -1 lt x lt 0 or x gt 1),(0, x=0, 1,-1),(1, 0 lt x lt 1 or x lt -1):}`
D. `fog(x)={{:(-1 , x lt 0 or x gt 1),( 0, x =0, 1,-1),(1, 0 lt x lt 1 or x lt -1):}`

Answer

To solve the problem, we need to evaluate \( f(g(x)) \) given the piecewise function \( f(x) \) and the function \( g(x) \).

### Step 1: Define the functions
The function \( f(x) \) is defined as:
- \( f(x) = -1 \) for \( x < 0 \)
- \( f(x) = 0 \) for \( x = 0 \)
- \( f(x) = 1 \) for \( x > 0 \)

The function \( g(x) \) is defined as:
\[ g(x) = x(1 - x^2) \]

### Step 2: Analyze \( g(x) \)
We can rewrite \( g(x) \):
\[ g(x) = x(1 - x^2) = x - x^3 \]

### Step 3: Find the roots of \( g(x) \)
To find where \( g(x) = 0 \):
\[ g(x) = 0 \Rightarrow x(1 - x^2) = 0 \]
This gives us the roots:
- \( x = 0 \)
- \( 1 - x^2 = 0 \Rightarrow x = \pm 1 \)

Thus, the roots of \( g(x) \) are \( x = -1, 0, 1 \).

### Step 4: Determine the intervals for \( g(x) \)
Next, we analyze the sign of \( g(x) \) in the intervals defined by its roots:
1. For \( x < -1 \): \( g(x) < 0 \)
2. For \( -1 < x < 0 \): \( g(x) > 0 \)
3. For \( 0 < x < 1 \): \( g(x) > 0 \)
4. For \( x > 1 \): \( g(x) < 0 \)

### Step 5: Evaluate \( f(g(x)) \)
Now we can evaluate \( f(g(x)) \) based on the intervals:
- **For \( x < -1 \)**: \( g(x) < 0 \) so \( f(g(x)) = -1 \)
- **For \( -1 < x < 0 \)**: \( g(x) > 0 \) so \( f(g(x)) = 1 \)
- **For \( x = 0 \)**: \( g(0) = 0 \) so \( f(g(0)) = 0 \)
- **For \( 0 < x < 1 \)**: \( g(x) > 0 \) so \( f(g(x)) = 1 \)
- **For \( x > 1 \)**: \( g(x) < 0 \) so \( f(g(x)) = -1 \)

### Step 6: Summarize the results
The final result for \( f(g(x)) \) can be summarized as:
- \( f(g(x)) = -1 \) for \( x < -1 \) and \( x > 1 \)
- \( f(g(x)) = 1 \) for \( -1 < x < 0 \) and \( 0 < x < 1 \)
- \( f(g(0)) = 0 \)

### Conclusion
Thus, the piecewise function \( f(g(x)) \) can be expressed as:
\[ f(g(x)) = \begin{cases} -1 & \text{if } x < -1 \text{ or } x > 1 \\ 1 & \text{if } -1 < x < 0 \text{ or } 0 < x < 1 \\ 0 & \text{if } x = 0 \end{cases} \]