When writing a function whose graph represents a combination of transformations, is the order not important?

Answer 1

depends on the transformation.

e.g. suppose we have f: (x,y) ----> (x,-y) , a reflection in the x-axis , and
g: (x,y) ---> (-x,-y) , a reflection in the origin

if we do fg
(x,y) ---> (x,-y) ---> (-x,y)
and if we do gf
(x,y) = (-x,-y) ---> (-x,y) , we get the same result, so for that combination the order does not matter,

but,

if u: (x,y) --- (x+2, y-1), a translation
and v: (x,y) ---> (-x,-y) , a reflection in the origin

uv ---(x,y) ---> (x+2,y-1) ---> (-x-2, -y+1)
vu ---(x,y) ---> (-x,-y) ---> (-x+2, y+1) , which is totally different
So in the last case, the order does matter