Intersections between two equations

Question

I am attempting to program a graphing calculator. I want to find the intersections between graphs on my program. Is it possible to find the intersection between two graphs given by $y=f(x)$ and $x=g(y)$ if it is only possible to evaluate either of the functions $f,g$ for arbitrary $x,y$? I know that I can solve for their intersection by using $y=f(x)=f(g(y)) \Rightarrow y-f(g(y))= h(y) = 0$ and then find the roots of $h(y)$, but is it possible to somehow find their intersection algorithically by evaluating $f(x)$ and $g(y)$ at different points?

Answer 1

Yes, it is possible, and the graph clearly indicates how you can proceed.

$$ x_{\,0} \; \to \;f(x_{\,0} ) = y_{\,0} \; \to \;g(y_{\,0} ) = x_{\,1} \cdots $$

The problem is to determine if the two graphs do have a cross point, or if they have more than one.
If they have more than one, you shall be able to isolate them into separated ranges and appropriately choose the starting point and axis.