difference between multiplying y by a coefficient and x by a coefficient

Hello,

So lets say we had the equation $\displaystyle g(x)=3(x+1)^2$

this is basically the parent function, $\displaystyle f(x)=x^2$, but shifted to the left 1 unit and expanded vertically by a factor of 3. The coefficient 3 is essentially being multiplied by f(x) aka y which could be represented by the expression $\displaystyle 3\cdot f(x)$. Since what is done to one side of an equation must be done to the other side, we must multiply (x^2) by 3 as well, or in this case, $\displaystyle 3(x+1)^2$.

So, i feel like i understand this. What i don't understand is, why when we are multiplying f(x) by 3 we say "...by a factor of three" and when we multiply x by 3 we say "...by a factor of one third." I don't see where the 1/3 comes from algebraically and it isn't explained in the book i am reading, it just expects the reader to accept it as a fact. Well, i accept it, but i still want to know where it comes from.

Thanks!