If `f:R to R` 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 1: Write down the function
We start with the function:
\[ f(x) = (3 - x^3)^{1/3} \]

### Step 2: Find \( f(f(x)) \)
To find \( f(f(x)) \), we substitute \( f(x) \) into itself:
\[ f(f(x)) = f((3 - x^3)^{1/3}) \]

### Step 3: Substitute into the function
Now we replace \( x \) in \( f(x) \) with \( (3 - x^3)^{1/3} \):
\[ f(f(x)) = \left(3 - \left((3 - x^3)^{1/3}\right)^3\right)^{1/3} \]

### Step 4: Simplify the expression
Calculating \( \left((3 - x^3)^{1/3}\right)^3 \) gives us \( 3 - x^3 \). Thus, we have:
\[ f(f(x)) = \left(3 - (3 - x^3)\right)^{1/3} \]

### Step 5: Simplify further
This simplifies to:
\[ f(f(x)) = \left(x^3\right)^{1/3} \]

### Step 6: Final simplification
Taking the cube root of \( x^3 \) gives us:
\[ f(f(x)) = x \]

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