Prove that \(f(f(x))=x\)
Question:
If
$$
f(x)=(a-x^n)^{1/n},
$$
where
$$
a>0 \text{ and } n\in\mathbb N,
$$
then prove that
$$
f(f(x))=x
$$
for all \(x\).
Solution
Given: $$ f(x)=(a-x^n)^{1/n} $$
Now, $$ f(f(x)) = \left[a-\left((a-x^n)^{1/n}\right)^n\right]^{1/n} $$
Since $$ \left((a-x^n)^{1/n}\right)^n=a-x^n $$
Therefore, $$ f(f(x)) = \left[a-(a-x^n)\right]^{1/n} $$
$$ = (x^n)^{1/n} $$
$$ =x $$
Hence, $$ \boxed{f(f(x))=x} $$