Is there no way to make this function?

Question

I want a function that curves similarly to this graph. Following the one equal to 1.

Basically I want the y axis minimum to be 0.5 and the y maximum to be 3. The x axis minimum to be 0 and the x maximum to be 2,500,000. So when I put 0 in the function it will be 0.5. This could be done linearly but I want a curve like the one in the graph above. I want the y axis to increase fast and then slow done.

I've asked this same question before but I didn't get help on it. I hope this is worded better for solutions. And yes I've looked into Cumulative distribution functions like the one in the picture but have no idea how to implement it like I want to.

Answer 1

Try with: $$y=0.5+2.5\left(\frac{x}{2500000}\right)^{\frac{1}{k\cdot\lambda}}$$

You can adjust the parameter $k\geq1$ in order to have the best fitting you need.

Answer 2

$f(x)=1-(x+1)^{-\gamma}$, should do the trick as a base, you can use linear transformations to make it your own