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

Step 1: Understanding the Concept:
Compute \(f(f(x))\) by substitution.
Step 2: Detailed Explanation:
\(f(x) = (a - x^n)^{1/n}\)
\(f(f(x)) = [a - (f(x))^n]^{1/n} = [a - (a - x^n)]^{1/n} = (x^n)^{1/n} = x\)
Step 3: Final Answer:
\(f[f(x)] = x\).