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Topic: Simple analytical properties of n/d
Replies: 20   Last Post: Mar 11, 2013 11:01 PM

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William Elliot

Posts: 1,618
Registered: 1/8/12
Re: Simple analytical properties of n/d
Posted: Mar 7, 2013 3:38 AM
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On Wed, 6 Mar 2013, Ross A. Finlayson wrote:
> > Exercise. Graph f:(R^2 - Rx{0}) -> R, (x,y) -> x/y.

> Mapping R^2 \ (0,0) the pointed disc to x/y, it is a many-to-one
> function where for each x =/= 0, for each r =/= 0 there is y s.t. for
> each x/y = r. So its image is the four quadrants minus the axes.


What's a pointed disk? R^2\(0,0) is neither a disk nor pointed; it's
a punctured plain.
No. The image of f is R.
R^2 - both axis is not the image nor the graph.
It's not even the projection of the graph onto the xy-plain.
. . which is R^2 - the y-axis.
The graph is a surface within xyz 3-space.

> What then of each r/y or x/r, as r ranges?

g(x) = x/a is linear; h(x) = b/x is hyperbolic.





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