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

[thenewmans]

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.

 

[I like Serena]

Greetings thenewmans!

Here's a non-trig solution:
$$y=\frac {\sqrt{16 x^2+9}-3} {2 x}+1$$

 

[thenewmans]

Perfect! Wow! Thank you!

 

[I like Serena]

Just for fun, here's another one that is closer to what you came up with.
$$y = \frac {2x}{\sqrt{x^2+3}}+1$$

 

[thenewmans]

OK, just tried it and it's even better because the slope at x=0 is closer to 1 (45 degrees). Thanks again!

 

[I like Serena]

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