Suppose `a gt 0` and `n in N` is odd.
Let `f : R to R` be defined by `f(x) = (a-x^(n))^(1//n)`, then `f(f(x))` is equal to:

Options

A. `nx`
B. `1/n x`
C. x
D. `|x|`

Answer

To solve the problem, we need to find \( f(f(x)) \) where the function \( f \) is defined as:

\[ f(x) = (a - x^n)^{\frac{1}{n}} \]

Given that \( a > 0 \) and \( n \) is an odd natural number, we can proceed with the following steps:

### Step 1: Find \( f(f(x)) \)

We start by substituting \( f(x) \) into itself:

\[ f(f(x)) = f\left((a - x^n)^{\frac{1}{n}}\right) \]

### Step 2: Substitute \( f(x) \) into the function

Now, we replace \( x \) in the function \( f \) with \( (a - x^n)^{\frac{1}{n}} \):

\[ f(f(x)) = \left(a - \left((a - x^n)^{\frac{1}{n}}\right)^n\right)^{\frac{1}{n}} \]

### Step 3: Simplify the expression

Next, we simplify the expression inside the parentheses. Since \( n \) is odd, raising to the power of \( n \) and then taking the \( n \)-th root will cancel out:

\[ \left((a - x^n)^{\frac{1}{n}}\right)^n = a - x^n \]

Thus, we have:

\[ f(f(x)) = \left(a - (a - x^n)\right)^{\frac{1}{n}} \]

### Step 4: Further simplify

Now, simplifying the expression inside the parentheses:

\[ a - (a - x^n) = x^n \]

So we get:

\[ f(f(x)) = (x^n)^{\frac{1}{n}} \]

### Step 5: Final simplification

Taking the \( n \)-th root of \( x^n \):

\[ f(f(x)) = x \]

### Conclusion

Thus, we conclude that:

\[ f(f(x)) = x \]