I do not think that the inverse of f(x) = x + x3 can be stated in closed form.
In order to find the inverse you would have to solve a cubic equation,
Well, that isn't quite true...because in this case the solution of the cubic is not too complicated although I will not try to reproduce it here.
Let me mess around a little more and get back to you.
OK, I am wrong there is a closed form expression! Here it is
Let z=sqrt(81x2+12)
then y = [.5x+(1/18)z]1/3 + [.5x-(1/18)z]1/3
You should graph this equation and the original equation on the same axes to see that it is correct.
Where did this answer come from?
In the original equation interchange the roles of x and y so that you get
x=y3 + y, i.e. y3 + y - x = 0
Now look up Cardan's formula for the solution of the cubic equation and make the appropriate substitutions; you should get the answer I gave you above. Good luck!
