Write a function that gradually stops at a certain value

Question

I need a function that looks like this:

It starts off like a linear function, but as $a$ increases, $x$ slowly starts to stop at $b$. I'm guessing I should have an extra parameter that describes how smooth the curve is, perhaps in relation to $a$?

Since I want the function to start linearly, I guess I should probably have $a$ and then subtract a value that slowly increases as $a$ increases? But how do I constrain it to $b$?

Answer 1

There are many ways. A simple way is $ba\over{b+a}$