f(x) is not defined at x = 1.
For x ≥ 1 , f (x) =

|x−1|

(x−1)

=

(x−1)

(x−1)

= 1
For x < 1, f (x) =

|x−1|

(x−1)

=

−(x−1)

(x−1)

=

(1−x)

(x−1)

= - 1
Thus, range of the function is either -1 or 1 at all the points and is undefined at x = 1