the range of thefunctiony=[x2 ] -[x]2, x belongs to [0,2] where [.] denotes the greatest integer function

When x lies in [0,1), x^2 lies in [0,1)

So f(x)= 0-0^2=0

When x lies in [1,√2), x^2 lies in [1,2)

So f(x)= 1-1^2=0

When x lies in [√2,√3), x^2 lies in [2,3)

So f(x)= 2-1^2=1

When x lies in [√3,2), x^2 lies in [3,4)

So f(x)= 3-1^2=2

When x=2, f(x)= 4-2^2=0

Hence range of f(x) is {0,1,2}

Dear student 
Ans given by Aditya is not standard.
Let x = t+1/2 
Then 
[x] = t 
[x^2] = [t^2 + 1/4 +t] = t^2 + t
[x2]−[x]2=t
As, t is an arbitrary integer, the range of the function is Z.