Question

Given that $f\left(x\right)={\left(a-x^n\right)}^{\frac{1}{n}};2>0,n\in N$, show that $f\left(f\left(x\right)\right)=x$.

Answer 1

$\begin{array}{c}f\left(x\right)={\left(a-x^n\right)}^{\frac{1}{n}}\\ f\left(f\left(x\right)\right)={\left(a-{\left({\left(a-x^n\right)}^{\frac{1}{n}}\right)}^n\right)}^{\frac{1}{n}}\\ ={\left(a-\left(a-x^n\right)\right)}^{\frac{1}{n}}={\left(x^n\right)}^{\frac{1}{n}}=x\\ \therefore f\left(f\left(x\right)\right)=x\end{array}$