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)`

Answer

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

### Step-by-step Solution:

1. **Identify the function**:
We have \( f(x) = (a - x^n)^{\frac{1}{n}} \).

2. **Find \( f(f(x)) \)**:
We need to compute \( f(f(x)) \). This means we will substitute \( f(x) \) into the function \( f \):
\[ f(f(x)) = f((a - x^n)^{\frac{1}{n}}) \]

3. **Substitute \( f(x) \) into \( f \)**:
Now, 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. **Simplify the expression**:
The term \( \left( (a - x^n)^{\frac{1}{n}} \right)^n \) simplifies to \( a - x^n \):
\[ f(f(x)) = \left( a - (a - x^n) \right)^{\frac{1}{n}} \]

5. **Further simplify**:
Now, simplifying \( a - (a - x^n) \):
\[ f(f(x)) = \left( x^n \right)^{\frac{1}{n}} \]

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

### Conclusion:
Thus, the final result is:
\[ f(f(x)) = x \]