I want to create an easing function $f(x,h)$ that looks like this:
I would like it to grow fast (so that $f(x,h) = h$ when $x$ is around $0.3$ and then it can fall slowly until $f(1,h) = 1$ (whatever $h$ is).
How can I build such function? I've been trying to approximate some functions but I cannot think of a way to build this without using recursion and splitting the function in subfunctions for $x \le 0.3$ and $x > 0.3$.
I'm guessing that when you say "easing function", you mean "the simplest possible". Assuming a maximum at $0.3$, a nice function with few terms that I could find is by imposing $$\begin{cases} f'(0.3)=0 \\ f(1)=1 \\ f'(1)=0 \end{cases}$$ in the function $y=e^{ax^2+bx+c}x^d$. In this way you will find $$y=x^{\frac{3}{5}a}e^{ax^2-\frac{13}{5}ax+\frac{8}{5}a}$$ and the relation between your $h$ and $a$ is $$(\frac{3}{10})^{\frac{3}{5}a}e^{\frac{91}{100}a}=h$$