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how to determine whether the graph of xy=3 is symmetric with respect to the x axis, y axis, y=x, y=-x

To be symmetric about the x axis, that means for any point (x,y) that is used,
the point (x,-y) must also be used.  This means it can't be a function for there
is only one point involved for each x in the definition of a function.
The function is really y = 3/x.  Since it can be defined as a function,
it is not symmetric about the x axis.

To be symmetric around the y axis, that means the function is even.
It says that for any y value defined at x, the same y value is defined at -x.
If the function has x values in it, they must be each to an even power.
since the x here is raised to the -1, and that is odd, it is not reflective about the y axis.

To be symmetric around the line y = -x, it would mean for any point (x,y) on the graph,
the point (-x,-y) would also be on the graph.  As can be seen, if xy = 3, then (-x)(-y) = 3
since (-x)(-y) = (-1x)(-1y) = (-1)(-1)(xy) = (+1)xy = xy.


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