possible transformations with upper and lower bound

Question

I have a given $$x\in (0,1)$$ and want to transform it with a ratio $$r\in [0,\inf)$$ and some kind of transforming function $f(x, r)$ so that the resulting value lies between a pre defined upper and lower bound, which are approached for very low or high ratios $r$: $$f(x,r) \in (x_u,x_o)$$ $$x_u, x_o \in (0,1)$$ $$x_u<x$$ $$x_o>x$$ $$f(x,1)=x $$ $$r\rightarrow 0 \Longrightarrow f(x,r) \rightarrow x_o $$ $$r\rightarrow \inf \Longrightarrow f(x,r) \rightarrow x_u $$ The lower the ratio, the higher the transformed value should be, and vice versa. What possible functions or transformations can you think of? How would you approach the process of finding such a transformation with the given properties?