The inverse function could be represented as an infinite series.
From $$y=x+\sin x$$ the inverse function satisfies, $$ y+\sin y = x $$
Let $$y=y(0)+ y'(0)x + y''(0) x^2/2+.....$$
We can find the derivatives using the equation $$ y+\sin y = x $$
We have $y(0)=0$.
Differentiation of $$ y+\sin y = x $$ implies $$ y'+\cos y y'=1$$
Evaluating at $x=0$, we have $ y'(0)=1/2 $
Similarly we can find higher derivatives and find the power series for $y$.