Polynomial/Degree 3/Inverse function/Power series approach/Exercise

Let

${}f(x)=-3x+x^{3}\,.$

Because of

${}f'(x)=-3+3x^{2}\,,$

this function is on the open interval ${}]-1,1[$ strictly decreasing and therefore injective (with the image interval ${}]-2,2[$). Also, ${}f(0)=0$. Let

${}g(y)=\sum _{k=0}^{\infty }b_{k}y^{k}\,$

be the inverse function, which we want to understand as a power series. Determine from the condition

${}(g(f(x))=x\,,$

the coefficients ${}b_{0},b_{1},b_{2},b_{3},b_{4}$.

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