I was wondering if someone could provide me with an intuitive answer for horizontal function transformation compressions.

Question

 Ex: expressing g(x) in terms of f(x). If you did f(2x) why would the x-coordinates be halved instead of getting bigger. That's literally the exact opposite of what it says is happening in those parenthesis

Answer 1

If I understand your question correctly. if g(x), f(x) and f(2x) are linear functions y values will be doubled for corresponding x values, thereby making the graph steeper. If you want to make graph flatter then f(x/2) would make it flatter.