Help creating a mystery function with these properties

Hello /sci/, basically I'm looking for a function that

>has f(0)=0

>for some integer m , has f(m)=0.5

>is bounded from above by 1

>is roughly this shape

how would i go about getting a function like this? the basic motivation is that i have lots of data points from [0, infinity] with a median of m and i want to assign scores from [0,1) to each of these data points.

let me know if i can give any more info to make the question more clear

Hello /sci/, basically I'm looking for a function that

>has f(0)=0

>for some integer m , has f(m)=0.5

>is bounded from above by 1

>is roughly this shape

how would i go about getting a function like this? the basic motivation is that i have lots of data points from [0, infinity] with a median of m and i want to assign scores from [0,1) to each of these data points.

let me know if i can give any more info to make the question more clear