firstly, if the cubic function is bijective, then of course it has an inverse. However, there is no direct relationship with the square term. Alternatively, you may check, if the original function is an increasing function, then definitely it has an inverse. You may use first derivative test to check it out. For example, X^3+x^2+4x+1=0, its first derivative is 3x^2+2x+4, and it's always positive, so, this function has an inverse although it has a square term.
Sorry, may have a lot of fallacies, pls enlighten me.:D
