If f (x) = [x], and g (x) = x-[x]. Find the ranges of f and g, and sketch the graph of g. Determine the solution sets of the equations: f (g (x)) = g (f (x))

I am not sure what you mean when you write [x], but let suppose [x] acts like a black box.

Then, f (x) = [x], range of f is any number satisfies that black box. For i. e: all complex and real number.

Same conclusion for g (x) = x-[x]

f (g (x)) = [x-[x]]

g (f (x)) = [x] - [[x]]

Here, without any detail (about the black box [ ]), then f (g (x)) = g (f (x)) with all x.