I am reviewing for a midterm for Pre-Calculus and I am trying to understand the concept of function transformation: Let's say I am given a function $f$ with the domain in the interval of $[1,5]$ and $g(x)=6-2f(x)$. Now my question is does it matter where you start your transformation? Can I move the graph up $6$ units then stretch it by a factor of $-2$? The textbook states to stretch it by a factor of $-2$ then move it up $6$ units. I tried both ways and ended up with different domains, $[-22,-14]$ and $[-4,4]$ respectively. So is there a certain order of operations to follow when transforming functions? ie: PEMDAS?
You are trying to grab at a rule where you should be trying to understand a concept. How do you get to $6-2f(x)$, starting from $f(x)$? Do you first multiply by $-2$, and then add 6? or do you first add 6 and then multiply by $-2$? What would happen if you took $f(x)$, and first you added 6, and then you multiplied by $-2$? What would you get?