What is the range of f(x) = lx-1l / (x-1) ? How do we calculate range?

The given function is ,
$f\left(x\right) =\frac{\left|x-1\right|}{x-1}$...........(1)
Domain of the given function will be all values of x except 1 and range of the function will be the different value of function at different value of the domain of the function ,
Since function is not defined at x = 1 so we can take two interval,
Case 1- when x < 1 , Then;
|x-1| = -(x-1) = 1-x now putting this value in equation (1) we get,
$\mathrm{f}\left(x\right) = \frac{\left|x-1\right|}{x-1}= \frac{1-x}{x-1}= -1$
Case 2 - when x > 1. then;
|x-1| = (x-1) now putting this value in equation (1) we get,
$\mathrm{f}\left(\mathrm{x}\right) = \frac{\left|\mathrm{x}-1\right|}{\mathrm{x}-1}=\frac{\mathrm{x}-1}{\mathrm{x}-1}=1$
So it is clear that whatever will be the value of the x , we will get 1 or -1 as the value of the function so
Range of the function will be 1 or -1. Answer
 
2) Range of a function can be found by first finding the domain of the function , after that find the maximum and minimum value of the function by putting the value of the domain and that would be the range of the function.