Find the inverse of 5throot of (x^5 + x^4).
Not much point in asking for the "inverse function" if you do not know if there is an inverse!
Perhaps you meant $\displaystyle f(x)= \sqrt[5]{x^5+ x^4}$ with a restricted domain such as $\displaystyle x\ge 0$?
As who listed? If it just said "find the inverse of the fifth root function", I would interpret that as "find the inverse of [tex]f(x)= x^{1/5}[tex] which is [tex]f^{-1}(x)= x^5[tex].
If it was "find the inverse function of the function '$\displaystyle \f(x)= \sqrt[5]{x^5+ x^4}$ for $\displaystyle x\ge 0$, Then would start by writing $\displaystyle x= (y^5+ y^4)^{1/2}$ then solve for y: $\displaystyle x^5= y^5+ y^5$ so $\displaystyle y^5+ y^4- x^5= 0$. Now, unfortunately, there is no simple "fifth degree equation formula" so there is no reasonable formula for solving that equation.
Ok. This precalculus textbook has many questions in the "Challenge" section at the end of each chapter. Personally, my answer would be: Question is too complex for this level of math. I would much rather find the inverse of functions like y = x^2 + 3 or
y = 1/(x+4) than to deal with fifth root functions.