Question

If f : R → R be defined by f(x) = (2 – x⁵)^1/5, then fof(x) = ___________.

Answer 1

Given: f(x) = (2 – x⁵)^1/5

$$\begin{gathered} fof\left(x\right)=f\left(f\left(x\right)\right) \\ =f\left({\left(2-x^5\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\right) \\ ={\left[2-{\left({\left(2-x^5\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\right)}^5\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.} \\ ={\left[2-{\left(2-x^5\right)}^{\frac{1}{5}\times 5}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.} \\ ={\left[2-2+x^5\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.} \\ ={\left[x^5\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.} \\ ={x^{5\times }}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.} \\ =x \end{gathered}$$


Hence, if f : R → R be defined by f(x) = (2 – x⁵)^1/5, then fof(x) = x.