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What are common functions that take two input variables and make the output between 0 and 1?

Question is as simple as that, two inputs and one output, output needs to stay between 0 and 1!

M.Mic
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    I think more information would be helpful...the constant $\frac 12$ works. As does $F(x,y)=\sin (xy)$ or $e^{-x^2}e^{-y^2}$. or $\frac 1{10x^2+173y^2+1}$. – lulu Jul 05 '17 at 12:21

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This question is way way way way way way too broad. There are infinitelly many such functions.

Examples:

  • $f(x,y)=|\sin(x)|$
  • $f(x,y)=\sin^2(x)\sin^2(y)$
  • $f(x,y)=1$
  • $f(x,y) = e^{-x^2-y^2}$

There are many more. Without further details, it's hard to give a more accurate answer.

5xum
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    Indeed, given any function $f(x,y)$ the function $g(x,y)=\dfrac{|f(x,y)|}{|f(x,y)|+c}$ satisfies the conditions for any positive constant $c.$ – mfl Jul 05 '17 at 12:26
  • Two inputs are have domains between 0 and 2. I have a more detailed version on my other questions but wanted to get a broad sense since that was not getting much response. Thank you! – M.Mic Jul 05 '17 at 12:27
  • Note that for any functions valued in $(0,1)$ you get another function valued in $(0,1)$ by multiplying them. – Theta Jul 05 '17 at 12:53
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$$f (x,y)=(x+y)-\lfloor x+y \rfloor $$

hamam_Abdallah
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For any lower and upper bounded function $f$, say:

$$m \le f(x,y) \le M$$

then the "normalized" function $g$:

$$g(x,y) = \frac{f(x,y)-m}{M-m}$$

will do the job. So any common bounded function can be scaled very simply to satisfy the condition. If you don't fear saturation, then:

$$g(x,y) = \min\left( \max\left(f(x,y),0\right),1\right)$$

does it as well. A useful class of such functions can be found among sigmoid functions (some take values in $[-1,1]$ but this can be fixed easily), like the logistic curve:

logistic curve

You also have a handful of cumulative distribution functions from probability distributions.

Laurent Duval
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