Building a function out of a given domain and range How does one build a function while given domain and range ? For example, the domain $(0,5]$ and range $[0, \infty)$

something like 
$f(x)= \frac 1x - \frac 15$ seems like it would fit the bill.
or $g(x) = -\ln x + \ln 5$
I am looking to map the open end in the domain with the open end in the range, and the closed end of the domain with the closed end of the range.  It is not entirely necessary, but if we don't it seems to me that we have to start looking for discontinuous functions.
$f(x)$ approaches infinity as x approaches 0.  And $f(5) = 0$
Any function with asymptotic behavior around $0$ should be a good place to start.  Then we do what we need to to fit the endpoint the other endpoint.  Functions that are strictly decreasing will be nice, to take away the possibility that they might dip below 0.  And continuous functions are nice, as they will go through every point between the extreme values of the range.
