If f: R R is defined by f(x)=[2 x]-2[x] for x R , then the range of f is (Here [x] denotes the greatest integer not exceeding x )

Step-by-step solution

Since, aligned x & =[x]+ x 2 x & =2[x]+2 x [2 x] & =2[x]+(2[x]) [2 x] & = cases 2[x]+0, & 0 < x < 1 2 2[x]+1, & 1 2 x < 1 cases [2 x]-2[x] & = cases 0, & 0 x < 1 2 1, & 1 2 x < 1 cases aligned Since, [2 x]-2[x]= cases 0, & 0 x < 1 2 1, & 1 2 x < 1 cases Hence, Range of f is 0,1