This is meant as a note. I will definitely expand this article later.

I hope this method has not been discovered before :(.

Say is a recursion relation. Then is the intersection point of and .

For example, if we have , then is the y-coordinate of the intersection of and .

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Nice post, but it will be more illuminating if you can give an example where solving the simultaneous equations $x=f(y)$ and $y=f(x)$ is easier than solving $x=f(x)$