Output shaping using a fixed input. I should know this :(

Hey guys.. So I'm working on a throttle management system for ebikes and I had a math questions for you fellas.

Okay so imagine you have an input we will call this X. X is a value from 750-4095

Right now currently the X value is taken and Y is output... Currently Y is from 750-4095

Y=X

WHAT I WANT TO DO:

I want to take in X (750-4095)

and output Y from 750-4095

But I want to add functions to the output.. For example I want exponential rise and exponential decay...

I realize it's as simple as using a function like Y=X^2 But that doesn't exactly do the trick.. I want my start value and end value to still be the same 750-4095...

For eg. I want the input to go from 750 to 1500 without making much of a change in the output say 750 -1000.. But then as the input goes from 1500-3000 have the output go from 1000-3000 with both input and output topping out at 4095

I hope this is clear.. I really should know the answer to this simple problem but I'm blanking out here.

Mo