Suppose we have a function $f:\mathbb{R}\to\mathbb{R}$ that is strictly decreasing, continuous, differentiable (twice), and convex.
Yet, we don't know the exact formula of $f$.
Can we approximate function $f$ by coming up with a surrogate function, say $g$? If yes, what are some standard ways to do this?
To me, it seems that, given our knowledge of $f$'s properties, one could show that there exists a function $g$ having the same properties with $f$ and is a scalar multiple of $f$, i.e., $f\approx\alpha\cdot g$, where $\alpha\in\mathbb{R}$. But I'm not sure if that's the case.
I'd appreciate any help (greatly appreciate any references). Thank you.