Comparing two functions with same trend

Question

Let $f$ and $g$ are two given functions with same trend. What I mean by same trend is following:

$$\forall x_1, x_2 \quad f(x_1) \leq f(x_2) \implies g(x_1) \leq g(x_2)$$ essentially, the above equation says that if $x_1$ and $x_2$ are two points such that $f(x_2)$ is greater than $f(x_1)$ then it is also the case that $g(x_2)$ is greater than $g(x_1)$.

If I know the maximum ($x_* | \forall x f(x) \leq f(x_*)$) for one of the functions (say $f$), is it always possible to bound the other function ($g$ in our case) by some simple transformation $T$ of $f(x_*)$ i.e.

$$if \quad f(x_*) = \max_x f(x)$$ $$is \quad \max_x g(x) \leq T(f(x_*)) \quad?$$ where $T$ is some transformation like translation.

In one dimension, I believe we can always translate $f$ to be always above $g$. To give more context, this comes in a multi fidelity setting where one of the function is an approximation of other where the two function values might not be of same scale but follow a "same trend" as defined above.

Answer 1

I don't know if I understood the question correctly, but it looks trivial to me. Let $f=0$ and $g$ be a constant. Then the hypothesis is satisfied. If $T(f{(x_{*})})$ any thing that depends only on $f$ but not on $g$ then we cannot expect $\max g(x) \leq T(F(x_{*}))$ since $g$ is an arbitrary constant.