Properties of inverse functions In order for a certain proof to work, I need a function and its inverse to satisfy the following property: if  $f(x) \ge k$, then $x \le f^{-1}(k)$. Is there a more general term for this? To provide some perspective, I am using $f$ to map distance to similarity. E.g. $1/(1+x)$ is one function that I am using for this purpose. So effectively the idea is that if the similarity is above some threshold, then the distance should also be below the same threshold, converted back to distance. 

Suppose $f$ is monotonically decreasing. Take some $x,y$ so that $f(x) > f(y)$ (you can call $f(y) = k$ and $y = f^{-1}(k)$). Then necessarily we have $x < y$. This shows that $f$ being monotonically decreasing is sufficient for the property you want.
If $f$ is invertible but not decreasing, then there is some $x,y$ so that $f(x) > f(y)$ but $x > y$. This shows that $f$ being decreasing is a necessary condition.
Note also that $f$ is invertible by assumption, so we cannot have $x \neq y$ with $f(x) = f(y)$. So your condition is equivalent to $f$ being monotonically decreasing. $\diamondsuit$
