So I am making an Arcade game and I am trying to make the difficulty curve smooth (now the values are hard coded), but I don't know how f(x) or f(g(x)) should look like in order to achieve this.
I found this resource Transforming Game Difficulty Curves usingFunction Composition , but I find it hard to understand as a whole. I have basic knowledge about functions and calculus but I haven't used them in a while.
Any advice or resource that I can research is well received.
I would love it if any potential solution to my problem comes with some explanation. I want to understand what I am doing 100% .
Thank you for your time and atentintion.
A typical function that models this sort of behavior is a logistic function: $$ f(x) = \frac{1}{1 + e^{-(x - a)/d}} $$ where $a$ is the "center" of the function (such that $f(a) = \frac12$), and $d$ is a parameter that controls the "steepness" of the transition from small values of $f$ to large values of $f$ (smaller $d$ means a "steeper" transition.)
Here's a Desmos calculator that allows you to play around with the parameters and see what works best for your application.