Question:
If f(x) = (a − xn)1/n, a > 0 and n ∈ N, then prove that f(f(x)) = x for all x.
Solution:
Given:
f(x) = (a − xn)1/n, a > 0
Now,
f{ f (x)} = f (a − xn)1/n
$=\left[a-\left\{\left(a-x^{n}\right)^{1 / m}\right\}^{n}\right]^{1 / n}$
$=\left[a-\left(a-x^{n}\right)\right]^{1 / n}$
$\left.=\left[a-a+x^{n}\right)\right]^{1 / n}=\left(x^{n}\right)^{1 / n}=x^{(n \times 1 / n)}=x$
Thus, f(f(x)) = x.
Hence proved.