How can I automatically scale this equation to a specific max $y$-level?

Question

I have a simple equation where:

• $f(x) = 1-e^{-x/a}$
• $g(x) = e^{-x/d}$
• $h(x) = f(x)g(x)$

This looks like this:

https://www.desmos.com/calculator/gquxkhxqci

I would like $h(x)$ to automatically scale to a max height of hitting $y=1$.

I note that it is possible to adjust the height of $h(x)$ by rewriting $g(x) = e^{h-x/d}$, where $h$ is a height constant. Could $h$ be automatically calculated or some other scaling method used to maintain a max of $y=1$?

Essentially I think I need a way of finding the $x$-value of $h(x)$ where $h(x)$ reaches its maximum $y$-level (ie. derivative = 0). Then I can divide $h(x)$ by this max $y$-level to scale it to that maximum.

How can this be done?

Thanks.

Answer 1

Never mind. I solved it with some help from Wolfram Alpha:

https://www.desmos.com/calculator/fvkz8o95pa