What is the inverse of the function

[kmarinas86]

I'm running out of ideas:

x=ay+by^3

Does someone here now how to solve for y in terms of x?

 

[Curious3141]

y^3 + \frac{a}{b}y - \frac{x}{b} = 0

Two ways :

1) Substitute y = z - \frac{a}{3bz}. Form a quadratic in z^3, solve for z and find y.

2) Compare equation to the trig identity cos^3 \theta - \frac{3}{4}\cos\theta - \frac{1}{4}\cos 3\theta = 0 while letting y = m\cos\theta then comparing coefficients. With this method, if you have to compute the arccosine of a value greater than one in magnitude, use the identity \cos i\theta = \cosh \theta

These are methods used to solve the general cubic in radicals/trig/hyperbolic trig ratios when the second degree term is missing (or has been eliminated).

 

[kmarinas86]

http://en.wikipedia.org/wiki/Cubic_equation

This helps too.