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How to mimic 4/pi*ArcTan(x)+1 without trig

  1. Aug 2, 2011 #1
    I need a basic math formula with the following properties:
    * limit y between -1 and 3.
    * (x, y) hits (-1, 0), (0, 1) and (1, 2).
    * Each y value occurs only once.

    I managed to do this with y=4/pi*ArcTan(x)+1. But I'd like to do this without trig. I got close with y=x*2/SQRT(1+x^2)+1. But it's not right. I keep thinking it's something simple and obvious. Can you help me?

    NOTE: This is not homework. I'm graphing and analyzing some proportion data in excel for myself.
     
  2. jcsd
  3. Aug 2, 2011 #2

    I like Serena

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    Greetings thenewmans! :smile:

    Here's a non-trig solution:
    [tex]y=\frac {\sqrt{16 x^2+9}-3} {2 x}+1[/tex]
     
  4. Aug 2, 2011 #3
    Perfect!!! Wow! Thank you!
     
  5. Aug 2, 2011 #4

    I like Serena

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    Just for fun, here's another one that is closer to what you came up with. :smile:
    [tex]y = \frac {2x}{\sqrt{x^2+3}}+1[/tex]
     
  6. Aug 3, 2011 #5
    OK, just tried it and it's even better because the slope at x=0 is closer to 1 (45 degrees). Thanks again!
     
  7. Aug 3, 2011 #6

    I like Serena

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    It's also better because the first one had a singularity at x=0, even though its limit was correct. ;)
     
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