How is it called if two functions have the same order?

Question

Lets have $f(x_1)>f(x_2)\implies g(x_1)>g(x_2) \forall x_i \in \mathbb{R}$. Is this property between $f$ and $g$ named in some way?

Answer 1

In the case $$f(x_1)>f(x_1)\Leftrightarrow g(x_1)>g(x_2)$$

for all $x_1,x_2\in \mathbb{R}$, we would say that $g$ is a monotonic transformation of $f$. This is because $g$ can be written as a composition $g=h\circ f$ where $h$ is a strictly increasing function.