Let $f: \mathbb{R_+}^n\to \mathbb{R_+}$ be a function that is strictly increasing in each of its arguments. Let $M_f$ be the set of its maximizers on some fixed compact subset $D\subseteq \mathbb{R_+}^n$, that is, $M_f := \arg \max_{x\in D} f(x)$.
Let $g$ be a function on $\mathbb{R}^n$ that multiplies each coordinate $i$ by a positive constant $a_i$, so that $g(x_1,\ldots,x_n) = (a_1 x_1,\ldots , a_n x_n)$.In general, the set of maximizers of $f\circ g$ is different than the set of maximizers of $f$. For example, if $f$ is the sum function, $f(x_1,\ldots,x_n) = x_1+\cdots+x_n$, then $(f\circ g)(x_1,\ldots,x_n) = a_1 x_1+\cdots+a_n x_n$, and the set of maximizers is clearly different. However, if $f$ is the product function, $f(x_1,\ldots,x_n) = x_1\cdot \cdots \cdot x_n$, then $(f\circ g)(x_1,\ldots,x_n) = (a_1\cdots a_n)\cdot (x_1\cdots x_n)$, and because all constants $a_i$ are positive, the set of maximizers of $f$ and $f\circ g$ is exactly the same.
Now, let $h$ be a function on $\mathbb{R}^n$ that adds a constant $b_i$ to each coordinate, so that $h(x_1,\ldots,x_n) = (x_1+b_1,\ldots , x_n+b_n)$.In this case, the set of maximizers of $f\circ h$ is equal to the set of maximizers of $f$ if $f$ is the sum function, but not if $f$ is the product function.
Finally, let $z$ be an affine transformation on $\mathbb{R}^n$, $z(x_1,\ldots,x_n) = (a_1 x_1+b_1,\ldots , a_n x_n+b_n)$, where $a_i$ are positive constants and $b_i$ are constants. Is there a function $f$ such that the set of maximizers of $f$ is equal to the set of maximizers of $f\circ z$, for any constants $a_i$ and $b_i$? The sum and product functions do not work, but maybe there is another function $f$ with this property?