Derivation of $[x + y] = [x] + [y + x - [x]]$ and the range of the $floor$ $function$

Question

One of the properties of the floor function is this : $[x + y] = [x] + [y + x - [x]]$

Please let me know what the derivation of this property is...

Another question that I have is : What is the range of the floor function? Is it $\Bbb Z$, the set of all integers? Please let me know

And, I also have a little 'less mathematical question' : Is the derivation of properties of functions, like this one important to be known and there are about 10 properties of the floor function given in my textbook, do I have to learn each one of those? I mean will those properties be used in further concepts or are those just a one time thing?

Thanks

Answer 1

We know $[x+n]=[x]+n$ for all n $\in Z$

So you RHS=$ [x]+[y+x-[x]]=[x]+[y+x]-[x]=[y+x]=LHS$ as [x] $\in Z$