real analysis - Bijection from $[0,1]$ to $(1, infty)$

I've come across many different versions of this question on here, but not any that map the $[0,1]$ to $(1, \infty)$.

I was thinking that it must be piece-wise defined, since the endpoints 0 and 1 will be the trickiest part of defining the bijection... The only method of doing this that I could come up with would be to possibly show a bijection from $[0,1]$ to $(1,2)$, then construct another bijection from $(1,2)$ to $(1, \infty)$, and then the composition will be from $[0,1]$ to $(1, \infty)$, but I haven't been able to come up with any function that can do this... Any help is much appreciated.