#f(x)# is polynomial, monotonic increasing. This can be stated from
#(df)/(dx)=3x^2+1>0 forall x in RR#
The inverse function #g(x)# is such that
#f(x)=y iff g(y) = x#
The inverse function calculation for
#f(x) = x^3+x =y#
involves the calculation of #x = g(f(x)) = g(y)# and this is obtained using an inversion formula such as Cardanos's
(see https://en.wikipedia.org/wiki/Cubic_function) for resolution of
#x^2+p x +q = 0#. Here #p = 1# and #q = -y#
#x = (-(q/2) + sqrt[(q/2)^2 + (p/3)^3] )^{1/3} +(-(q/2) -
sqrt[(q/2)^2 + (p/3)^3])^{1/3}#
giving
#x = (y/2 - sqrt[1/27 + y^2/4])^(1/3) + (y/2 + sqrt[1/27 + y^2/4])^(1/3)#
so
#
g(y) = (y/2 - sqrt[1/27 + y^2/4])^(1/3) + (y/2 + sqrt[1/27 + y^2/4])^(1/3)
#
The attached figure shows in blue #f(x)# and in red #g(x)#
