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From: sobriquet <dohduhdah@yahoo.com>
Newsgroups: sci.lang,sci.math
Subject: Re: f(x) = (x^2 + 1) --------- strange (curved Surface) Graph
Date: Tue, 30 Jul 2024 19:30:14 +0200
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Op 29/07/2024 om 21:28 schreef HenHanna:
> 
> When   this function      y =  f(x)  =  (x^2  +   1)      is first 
> introduced, we learn its Graph to be a  simple  parabola.
> 
> THEN  when we learn  that  x can be a complex number, so that
> the Graph  is    2 (orthogonally) linked   Parabolas.
> ---------- like this:
> 
> https://phantomgraphs.weebly.com/uploads/5/4/5/4/5454288/4_4_orig.jpg
> 
> https://www.geogebra.org/resource/czbugz9h/fofRh3ZjmwwISd2v/material-czbugz9h-thumb@l.png
> 
> 
> 
> This graph   is   showing a smooth ,  curved  surface   -->
> 
>                  https://i.sstatic.net/soSJ8.png
> 
> What is this graph showing???
> 
>                 it purports to show    f(x)  =  (x^2  +   1)

Here is one way to visualize it on desmos3d

https://www.desmos.com/3d/8tqp4wqzad

We can verify the plots with wolfram alpha (plotting re(f), im(f), 
abs(f), arg(f) respectively).

https://www.wolframalpha.com/input?i=plot+arg%28%28x%2Biy%29%5E2-1%29%2C+-5%3Cx%3C5%2C+-5%3Cy%3C5%2Cplotrange+%28-5%2C5%29

My function f is used to map the domain of 0 to 1 for parameters to the 
range -infinity to infinity.

The function g is used to multiply two complex numbers.