Question

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)]`

Answer 1

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.