Question 3
Let $\mathbb R_+$ denote the positive real numbers. Determine all functions $f:\mathbb R_+\to\mathbb R$ such that for all $x,y,z\in\mathbb R_+$,
$$|x-y|\lt |y-z|\quad\text{if and only if}\quad|f(x)-f(y)|\lt |f(y)-f(z)|.$$
$$|x-y|\lt |y-z|\quad\text{if and only if}\quad|f(x)-f(y)|\lt |f(y)-f(z)|.$$