On a graph you can limit the showing of a function between two x values by multiplying by a complex when outside.
$$\\f(x) = ax * \frac{(\sqrt{(max -x)} * \sqrt{(x - min)}}{(\sqrt{|(max -x)|} * \sqrt{|(x - min)|}}$$
I was wondering if it was possible to make $$\\f(x) = 1$$ instead when outside these ranges to make it possible to chain multiple functions together.
You have noted that the function $$ \frac{\sqrt{x-c}}{\sqrt{|x-c|}} $$ equals $1$ if $x>c$ and is undefined when $x\le c$. Consider the variant $$ g_c(x) = \frac12\bigg(\frac{|x-c|}{x-c}+1\bigg), $$ which equals $1$ if $x>c$ and $0$ if $x<c$ (and, unfortunately, is undefined when $x=c$). You can take any function $h(x)$ and change it to be $0$ outside some interval $[min,max]$ by setting $$ f(x) = h(x) g_{min}(x) (1-g_{max}(x)). $$ And this can be chained together using addition to allow for multiple ranges.