If `f(x)=(a-x^(n))^(1//n),"where a "gt 0" and "n in N`, then fof (x) is equal to

Options

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

Explanation

To solve the problem, we need to find \( f(f(x)) \) where \( f(x) = (a - x^n)^{\frac{1}{n}} \) and \( a > 0 \), \( n \in \mathbb{N} \).

### Step-by-Step Solution:

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

2. **Find \( f(f(x)) \)**:
To find \( f(f(x)) \), we need to substitute \( f(x) \) back into the function \( f \):
\[ f(f(x)) = f\left((a - x^n)^{\frac{1}{n}}\right) \]

3. **Substituting into the function**:
We replace \( x \) in \( f(x) \) with \( (a - x^n)^{\frac{1}{n}} \):
\[ f(f(x)) = \left(a - \left((a - x^n)^{\frac{1}{n}}\right)^n\right)^{\frac{1}{n}} \]

4. **Simplifying the expression**:
Now, we simplify \( \left((a - x^n)^{\frac{1}{n}}\right)^n \):
\[ \left((a - x^n)^{\frac{1}{n}}\right)^n = a - x^n \]
Thus, we have:
\[ f(f(x)) = \left(a - (a - x^n)\right)^{\frac{1}{n}} \]

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

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

### Final Answer:
The answer is \( f(f(x)) = x \).