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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?

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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?

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I think I understand how this works, taking $f(x)$ multiplying it by $-2$ then adding $6$ is what I want. But taking $f(x)$ and adding $6$ then multiplying by $-2$ would give me $-2f(x)-12$ is that correct? – Kot Oct 7 '12 at 22:49
Yes, that's correct. In the same way that you can't arbitrarily interchange the order of addition and multiplication, you can't arbitrarily interchange the order of translation (movement) and homothety (stretching). When it comes to formulas, those geometric transformations are simply addition and multiplication again :) – Yoni Rozenshein Oct 7 '12 at 23:19

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