I am confused about (dummy?) variable substitutions in comparison with change of variable type of substitutions.
On the one hand, I can write $f(x) = \dfrac{1}{1+x}$ or $f(u) = \dfrac{1}{1+u}$ and the functions produce same result for equal values of $x$ and $u$.
On the other hand, if $x$ and $u$ are related, say $x = 2u$, I get $f(x) = f(2u) = \dfrac{1}{1 + 2u}$. Now if I refer to $f(u)$, would I be talking about $\dfrac{1}{1 + u}$ or $\dfrac{1}{1 + 2u}$? Obviously in this case, $f(x=5)$, for example, would not product the same result as $f(u=5)$ without knowing which $f(\cdot)$ we are talking about.
Is there a more formal way to think about this? Should one of the functions $f(\cdot)$ actually be called something else?
When you write $f(x) = \frac{1}{1+x}$ you are referring to the function $f\colon x\mapsto \frac{1}{1+x}$. Writing $f(u)$ means the same thing, except you've relabeled your variable. If you make the substitution $x = 2u$ so that you have $f(x) = f(2u) = \frac{1}{1+2u}$, you are still referring to the same function $f$, but you are inputting $2u$. If you want to refer to the function $u\mapsto \frac{1}{1+2u}$ in the same context then you should call it something other than $f$.