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The figure on the left shows a family of parabolas: . y = x² + c

Where 'c', the "numerical coefficient", is varied from -2, (cyan), through 0, (blue), to 2, (heavy black)

The effect of the numerical coefficient is simply to raise or lower the entire function, the shape does not change.

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Mathematically, the figure on the right is more interesting -- it shows the inverses of the functions on the left.

The heavy black 1 / ( x² + 2 ) peaks at 1/2, since its parent function on the left, x² + 2, bottoms at 2.

Then, as the functions at left are lowered towards 0, where the blue function bottoms,

their inverses climb to infinity, where the blue function at right will top out.

When the functions at left bottom below 0, their inverses at right travel "around the globe", to come up from below.

The lowest function at left, in cyan, bottoms at -2. .Its inverse at right, has a relative maximum at -1/2.

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There are two ways of looking at this, one is that the co-ordinate plane is actually a sphere.

The other is that division by 0 is undefinable, and therefore the functions have an indeterminate value as they jump across the plane.

The second way is simpler to use, it keeps things flat and preserves the nullity of zero.

But looking at the graphs, the first way makes more sense.

For one thing it allows the functions on the right to be as continuous as those on the left. .(Continuity is important to math).

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Secondly: .Families of functions respect one-another. .If they cross at all, they will do so together, at gatherings.

( See the family roots on the page: .The W )

You shouldn't be able to modify 'c' so that the new curve will cross over the old one in an incidental manner.

But if inverse functions magically vault the number plane when their input is zero, they also magically cross their brothers and sisters.

And each does so at a place and time of its own choosing, so there is no family gathering.

Although this doesn't produce "wrong" results, (there is no result to be wrong), it is poor philosophically.

That is covered in more detail on the page: .On Zero, Infinity and One .