Trying to understand at a deeper level what graph transformations are doing to the original function.
I found this answer to almost get me there, but some parts are confusing: https://math.stackexchange.com/a/2391593/80778
Some questions are about semantic reasoning.
If we perform the substitution $x \rightarrow x+1$ on $f(x,y) = y - f(x) = 0$ resulting in $y - f(x+1) = 0$, is this defining a new function $f(x,y)$? Function transformations have graphs $G'$ which are different from their parent functions, should we think of them as entirely separate functions?
Well, this new function after transformation is composed of points $(x-1,y)$. I can reason that graphically. I think the answer I cited is explaining the algebraic reason for this, but either the notation is not clear enough (I'm being nitpicky I'm sure) or I'm just not understanding.