Looking for an unbounded, monotonic, non-symmetric s-shaped function

I am hoping to find a function that corresponds to an s-shaped curve that satisfies the following properties: 1. unbounded 2. it is not symmetric about its inflection point 3. its derivative at the inflection point is not infinity 4. it is monotonic

Unfortunately, none of the many sigmoid functions listed below satisfy the first two of my requirements

https://en.wikipedia.org/wiki/Activation_function

I would be very grateful for any suggestions. Thank you in advance!

• Sure you want unbounded instead of bounded? – Michael Hoppe Apr 28 at 8:38
• No one is going to read through a research paper to see what you are talking about. Provide all the information you think people might require in your question. – Paul Apr 28 at 8:45
• Yes - bounded. Those links just provide examples of s-curves that the researchers (especially the people working with neural networks) have been using. All of them seem to be bounded and symmetrical. Isn't there Any function out there that isn't? – BillB Apr 28 at 16:48
• Something like $g(x) = af(x/a)$ when $x>0$ and $g(x)=f(x)$ when $x \le 0$ should create the asymmetry for any of the smooth activation functions in the wikipedia page. You can probably find something to multiply by on the right to make the function unbounded, if that's really what you want. – Ethan Bolker Apr 28 at 17:13
• Thank you, Ethan. Your advice about introducing asymmetry helps a lot! I have been trying and failing to find something to multiply by (or another transformation) to introduce unboundedness... – BillB Apr 28 at 20:20