Let
-
![{\displaystyle {}f(x)=-3x+x^{3}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad27254ec7d221936f74465aedde1d481f305960)
Because of
-
![{\displaystyle {}f'(x)=-3+3x^{2}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ecad2c454c19c1963ede6288a5001255923b881)
this function is on the open interval
strictly decreasing and therefore injective
(with the image interval
).
Also,
.
Let
-
![{\displaystyle {}g(y)=\sum _{k=0}^{\infty }b_{k}y^{k}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17d98176326c758b7a6c2ced492caff20eeff16d)
be the inverse function, which we want to understand as a power series. Determine from the condition
-
![{\displaystyle {}(g(f(x))=x\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69696258c3e817e2e0f276cae289661344004811)
the coefficients
![{\displaystyle {}b_{0},b_{1},b_{2},b_{3},b_{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1011968a4aa27855a80194d377dd82eb218b5df0)
.