Question

Let $f\left(x\right)=(a-x^n{)}^{\frac{1}{n}}$,$a>0$ and $n$ is positive integer, then prove that $f\left[f\right(x\left)\right]=x$.

Answer 1

Given $f\left(x\right)=(a-x^n{)}^{\frac{1}{n}}$ for $a>0$ and $n$ is positive integer.

Now

$f\left[f\right(x\left)\right]$

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

$=[a-a+x^n{]}^{\frac{1}{n}}=x$