Convert Bézier curve to equation

How to convert for example this Bézier curve: cubic-bezier(.65,0,.65,1) (plot) to an equation like f(x) = x... ?

You can't; not easily, anyway. The way the app works is that the four numbers $(a,b,c,d)$ represent the cubic Bezier curve with control points $\mathbf{P}_0 = (0,0)$, $\mathbf{P}_1 = (a,b)$, $\mathbf{P}_2 = (c,d)$, $\mathbf{P}_3 = (1,1)$.
This curve can be written in parametric form as $$x(t) = 3 a t + (3c -6 a) t^2 + (1 + 3 a - 3 c) t^3$$ $$y(t) = 3 b t + (3d -6 b) t^2 + (1 + 3 b - 3 d) t^3$$ If you want to write this in the form $y = f(x)$, then the "$f$" would have to include all the algebra for solving a general cubic equation, which is a rather nasty mess.