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

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

1 Answer
Apr 23, 2018

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

Explanation:

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.