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This is just geometry (and functional analysis). You're dealing
with a function that has two branches. The same thing occurs if you
consider integrating the differential equation for a circle:
d/dx(y^2) =-2x
there is a solution for positive y and a solution for negative y.
The two solutions together map the whole circle.
Since the solution is y = sqrt{const - x^2} it is obvious
from the square root that my solution must have two distinct
branches, which immediately tells me something about the physics.
In your case that something is: going up ain't the same as going
down.
Regards,
Jack