If `f:RrarrR` be given by `f(x)=(3-x^(3))^(1//3)`, then fof(x) is :

Options

A. `x^(1//3)`
B. `x^(3)`
C. x
D. `3-x^(3)`

Answer

To solve the problem, we need to find \( f(f(x)) \) where \( f(x) = (3 - x^3)^{1/3} \).

### Step-by-Step Solution:

1. **Define the function**:
We start with the function given:
\[ f(x) = (3 - x^3)^{1/3} \]

2. **Substitute \( f(x) \) into itself**:
We need to find \( f(f(x)) \). This means we will replace \( x \) in the function \( f \) with \( f(x) \):
\[ f(f(x)) = f((3 - x^3)^{1/3}) \]

3. **Calculate \( f(f(x)) \)**:
Now we substitute \( (3 - x^3)^{1/3} \) into the function:
\[ f(f(x)) = \left( 3 - \left( (3 - x^3)^{1/3} \right)^3 \right)^{1/3} \]

4. **Simplify the expression**:
The cube of \( (3 - x^3)^{1/3} \) is simply \( 3 - x^3 \):
\[ f(f(x)) = \left( 3 - (3 - x^3) \right)^{1/3} \]
Simplifying further:
\[ f(f(x)) = \left( x^3 \right)^{1/3} \]

5. **Final simplification**:
The cube root of \( x^3 \) is \( x \):
\[ f(f(x)) = x \]

### Conclusion:
Thus, we have found that:
\[ f(f(x)) = x \]