Question

If \(f(x) = (a - x^n)^{1/n}\), where \(a>0\) and \(n\) is a positive integer, then \(f[f(x)]\) is equal to

• A. \(x^3\)
• B. \(x^2\)
• C. \(x\)
• D. None of these

Hint:

\(f(x)\) is the inverse function of itself (involution).

Answer 1

The Correct Option is C

To find \( f[f(x)] \) for the function \( f(x) = (a - x^n)^{1/n} \), let's tackle this problem step-by-step:

1. First, identify \( f(x) \):
   \(f(x) = (a - x^n)^{1/n}\)
2. Substitute \( f(x) \) into itself to find \( f[f(x)] \):
    

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

1. Calculate \( f\left((a - x^n)^{1/n}\right) \):
   Plugging the expression from step 2 into \( f \), we have:
   \(f\left((a - x^n)^{1/n}\right) = \left(a - \left((a - x^n)^{1/n}\right)^n\right)^{1/n}\)
   Notice that \(\left((a - x^n)^{1/n}\right)^n = a - x^n\).
2. Simplify the expression:
   \(\left(a - \left((a - x^n)^{1/n}\right)^n\right)^{1/n} = \left(a - (a - x^n)\right)^{1/n} = (x^n)^{1/n}\)
3. Simplifying further gives: \((x^n)^{1/n} = x\)
   This means that the expression reduces back to \( x \).
4. Therefore, \( f[f(x)] = x \).

Conclusion: Based on the above calculations, the correct answer is \( x \).