I am trying to come up with a function y=f(a) that includes a fixed parameter b such that the graph has a minimum at x=0, a maximum of 1 at x=b, and a right tail that approaches the x-axis asymptotically as x approaches infinity after x>b. I want the function to approximate the combinationof y=x/b where x<s, and y=b/x where x>b, but with a smooth curve. Any insight would be greatly appreciated. Thanks!

You may start with something like

fc(x) = (log(x+c) - log(c)) / (x + c)

This is a positive function (for x > 0) that aproaches 0 at infinity, is 0 at 0, and has a single maximum. This is true for any c > 0.

Now you can consider, say, n·fc(m·x) with three parameters to choose the most suitable function. For b, you must be able to choose one of the three or some combination thereof by relating their values.

Actually, the base of the logarithm is a fourth parameter.

Since you ask for a smooth curve perhaps you will not like to use a modulus. Otherwise the Cupid's Bow function Mod(2bx/(x^2+1))seems to satisfy your requirements