Hi,
The following is behavior is for Mathematica version 13.3.
Why is part of the region in the negative x range? Given the specification I would have expected a there would be a straight line from {0,0} to {1,1}, with no part of the region to the left of that line.
First off, in your picture the axis looks to be at {.5,0}, not {0,0}. Second, the generic display of regions can sometimes be inaccurate (I think this is due to sparse sampling).
You could try (I’ve added the origin for clarity)
RegionPlot[x >= y && x <= 1/y, {x, 0, 3}, {y, 0, 3}, AxesOrigin -> {0, 0}]
Or just discretizing your region gives a more accurate visualization.
DiscretizeRegion[r]