Question

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))

Answer 1

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.