If two variables $x$ and $y$ are both known to be in the domain of a unary function $f$ (but not necessarily defined over the whole domain nor be equal to the other; i.e., $A∋x,y$ while $◇\:\{x\}≠\{y\}\quad ⇔\quad ¬□\{y\}=\{x\}$), then could $y$ be replaced by $x$ and vice-versa respectively in the expressions “$f(x)=\:\text{. . }x\text{ . . }$” and “$f(y)=\:\text{. . }y\text{ . . }$” (Would equivalently-true statements result)? If so, then is there a correct way to state this symbolically? If not, then Why not and How might the description be tightened or loosened to ensure that it is true?
An example implication of (the validity or not of) this suggested de facto $f(\cdot)$-substitution is $f(x)=2x^2-5x+7 \:\:\stackrel{?}{⇔}\:\: f(y)=2y^2-5y+7$.
First of all, I think you mean that the variables $x$ and $y$ stand for elements of the domain of $f$, not that they are in the domain of $f$. Now, are you using those variables to stand for particular, fixed elements of the domain? Or are you using them as quantified variables ranging over the domain?
If $x$ and $y$ stand for particular elements of the domain, not necessarily the same element, then the statements $f(x) = 2x^2 - 5x + 7$ and $f(y) = 2y^2 - 5y + 7$ are not equivalent.
But sometimes people write equations like this with the intention that the variables are to be understood to be universally quantified. In other words, the equations are intended to mean $\forall x \in A(f(x) = 2x^2 - 5x + 7)$ and $\forall y \in A(f(y) = 2y^2 - 5y + 7)$. Those statements are equivalent.
This is a good illustration of the fact that if there is any possibility of confusion, it is best to write quantifiers explicitly.