How to find the formula for the inverse of f(x) given below?

Question

#f(x)=1/3x^3(4-x^2)^(1/2)# defined on #[0,3^(1/2)]#

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Answer 1 · 2018-04-23T05:15:51

This is equivalent to solving for #x# in # x^8 - 4x^6 + y^2 = 0# so no joy.

graph{(1/3) x^3 (4-x^2)^(1/2) [-5, 5, -2.5, 2.5]}

It looks like it has a well defined inverse at on that interval. You could even extend it to #[- sqrt{3}, sqrt{3}]#.

I have my doubts about being able to find a closed form formula for this inverse.

Solve for #x#:

#y = 1/3 x^3 (4-x^2)^{1/2}#

#y^2 = 1/9 x^6 (4-x^2) #

#9 y^2 = 4x^6 - x^8 #

# x^8 - 4x^6 + y^2 = 0#

I don't know how to solve an eight degree polynomial equation. Uncle.