Finding the range of a floor function without the graph?

Question

I don't have a problem with finding the domain and the range of any function however I get stuck when I see this sign [x]

f(x)=[x]+x

This function for example how I can find the range in proper method.

I thought that [X]<x<[x]+1 could help .

Answer 1

They way I would solve it is to look what are the values of this function on intervals $[n,n+1)$ separate and then sum all those ranges. We know that for $x\in[n,n+1)$, $f$ is given by $$ f(x) = n+x $$ and the range on this interval is $[2n,2n+1)$ (see that $f$ is continuous and increasing on the interval that we consider). We can denote this set as $A_n$ and then we have $$\bigcup_{n \in \mathbb{Z}} A_n = \bigcup_{n \in \mathbb{Z}} [2n,2n+1) = \dots [-2,-1) \cup [0,1) \cup [2,3) \dots $$ I'm not sure how to write this answer in more elegant way but you see the pattern.