I spent quite a bit of time tweaking this relation. (It's a relation and not a function because the "o" can't pass the vertical line test). Ultimately I settled for something relatively simple.
If you want to put this into your graphing calculator, I recommended:
Y1 = (1+(1-x*x)^(1/3))(abs(x) < 1)"abs()" can be found under MATH => NUM. The "<" can be found under TEST (2nd MATH) and the "and" is under TEST => LOGIC.
Y2 = (1-(1-x*x)^(1/3))(abs(x) < 1)
Y3= (1+cos(πx))(abs(x) < 5 and abs(x) < 1)
You get this weird vertical line thing at x = 1 or -1, but that's because the calculator has a hard time handling discontinuities and fills in the gap the best it can. In a perfect world, there would be a tiny gap between the bottom of m and the side of the o.
If you're not a fan of piecewise functions, then you have to do some magic with absolute values:
I know what you're thinking, absolute values are piecewise functions, you're not fixing the problem at all. Well, that's only if you define absolute value as:
If you define it as:
Then everything is good to go.
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