I am trying to create a function of A and x which has the following properties. A is a scaling parameter that determines the shape of the function. I write the function below in f(A,x) form 1) f(A,1)=1 always 2) For all x>1, 0<f ' (x)<1 3) As A approaches some upper bound (which could be infinity), the function approaches f(A,x)=x (i.e. the simple graph y=x) 4) As A approaches some lower bound (which could be infinity), the function approaches f(A,x)=1 (i.e. the simple graph y=1) 5) I only care about the shape of the graph for values of x greater than 1 (so, conditions 2, 3, and 4 don't need to hold for values lower than 1 The closest I've come is y=A*ln(e^(1/A)*x), for A between 0 and 1. This achieves conditions 1, 2 and 4, but not 3.