General method for reflection a graph over a nonlinear base?

I'm talking complex relfections here. Not a line reflected over any particular seperate line (especially not the simple x-axis, y-axis, or y=x reflections). My question concerns weather or not there is a known general method to reflecting a certain function over a curved function. I think i can do this point by point but its a pain in the *** to even do one point without running out of paper. Consider the equation f(x)=x^2 + 3 and the line g(x)= (1/3)(x). My process starts by finding the line normal to g(x) at a given point, than finding exactly where it intersects f(x). The next step is finding the angle between the normal line of g(x) and the line tangent to f(x) at the point of intersection. Than I construct a line through the same point with an equal angle in the opposite rotation from the line tangent to f(x). Finnally, I'd find the distance of my orginal normal of g(x) to the point of interesection on f(x) and than construct a point along the line that is the same distance from f(x), upon the line which was contructed at the same angle away from the tangent line. Now, my questions is this: other than sitting down and plotting a large number of these points until i could "visualize" the graph of g(x) that has been reflected over f(x) (lets call it R(x) ), is there a general form to this process that will result in the conclusion of this graph, described as a function of x? or maybe R(x) can be described parametrically? please please please. somebody help me out. The math forums have been no help and neither has my teacher. If you have any questions let me know, maybe I can scan my graphs and post them to make the process more clear

Check out Mobius transformations (also known as linear fractional transformations) of the complex plane. They reflect the entire complex plane across a circle (or a line, which is a circle with its center at infinity). This is done conformally (in a way that preserves angles, so that locally it looks like you're only scaling, rotating, and translating), which I think you want; otherwise, the image of your curve might not bear any resemblance whatsoever to what you started with.

The bad news is that if you need to conformally map the entire plane onto itself (e.g. one side of a curve to the other side, and conversely), then Mobius transformations are all you have to work with.