Find the inverse of $f(x) = 3x + \sin(\pi x)$

Question

I understand that to find the inverse you switch the $x$ and $y$ then solve for $y$.
However, in this case you end up with $x=3y + \sin(\pi y)$. I know that the inverse exists and is a function since the $f$ is monotonic for all $x$.
Is there any way for me to isolate the $y$ and solve for the inverse?

Answer 1

$$3x+\sin(\pi x)=y$$ $x\to\frac{t}{\pi}$: $$\frac{3}{\pi}t+\sin(t)=y$$ $$t+\frac{1}{3}\pi\sin(t)=\frac{1}{3}\pi y$$

This isKepler's equation.

see How to solve Kepler's equation $M=E-\varepsilon \sin E$ for $E$?