If `f(x) = (a-x^n)^(1/n)` then `fof(x)` is (A) x (B) a-x (C) `x^2` (D) `-1/x^n`

Options

A. a
B. x
C. `x^(n)`
D. `a^(n)`

Answer

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

### Step-by-Step Solution:

1. **Write down the function**:
\[ f(x) = (a - x^n)^{1/n} \]

2. **Find \( f(f(x)) \)**:
We need to substitute \( f(x) \) into itself:
\[ f(f(x)) = f((a - x^n)^{1/n}) \]

3. **Substitute \( f(x) \) into the function**:
Replace \( x \) in \( f(x) \) with \( (a - x^n)^{1/n} \):
\[ f(f(x)) = \left( a - \left( (a - x^n)^{1/n} \right)^n \right)^{1/n} \]

4. **Simplify the expression**:
The term \( \left( (a - x^n)^{1/n} \right)^n \) simplifies to \( a - x^n \):
\[ f(f(x)) = \left( a - (a - x^n) \right)^{1/n} \]

5. **Further simplification**:
This simplifies to:
\[ f(f(x)) = \left( x^n \right)^{1/n} \]

6. **Final simplification**:
The expression \( \left( x^n \right)^{1/n} \) simplifies to \( x \):
\[ f(f(x)) = x \]

### Conclusion:
Thus, we find that \( f(f(x)) = x \).

The correct answer is **(A) x**.