If f(x)=log[x−1]⁡|x|x, where [.] denotes the greatest integer function, then

f(x) is defined for all x satisfying[x−1]>0,[x−1]≠1 and |x|x>0.Now, [x−1]>0,[x−1]≠1⇒x∈[3,∞)and , |x|x>0⇒x>0∴ D(f)=[3,∞)Clearly, |x|x=1 for all x∈[3,∞)∴ f(x)=log[x−1]⁡|x|x=log[x−1]⁡1=0So, R(f)={0}