Notating function increase

Question

Is there a way to formally notate the following statement?

For a function $f(x)$, as $x$ increases, $f$ decreases, and as $x$ decreases, $f$ increases

Answer 1

There isn't symbolic notation that I'm aware of, but typically the standard term is "$f(x)$ is a decreasing function", with "$f(x)$ is a strictly decreasing function" if $f(x)$ never "levels off" and has zero gradient.

Answer 2

This sounds like order-reversing / antitone function from order theory point of view.