Why are horizontal transformations of functions reversed? While studying graph transformations I came across horizontal and vertical scale and translations of functions. I understand the ideas below.


*

*$f(x+a)$ - grouped with x, horizontal translation, inverse, x-coordinate shifts left, right for -a

*$f(ax)$ - grouped with x, horizontal scaling, inverse so x-coordinate * 1/a

*$f(x)$ + a - not grouped with x, vertical translation, shifts y-coordinate up, d

*$af(x)$ - not grouped with x, vertical scaling, y-coordinate * a


I have mostly memorized this part but I am unable to figure out why the horizontal transformations are reversed/inverse?
Thanks for your help.

For horizantal shift:
The logical reason for horizantal shift is that in(f)(x)=y=x the origin is (0,0)and in f(x)=(x-2)is (2,0) for this we should add 2 to get 0 becouse in parent function become 0 when we add 0 and in shifted function to make zero our function we ahould add 2
