Another way to find range is to "reverse" the fraction- solve for x- so that domain and range are swapped.
If then [tex]y(x-1)= xy- y= x. Then xy- x= y so x(y- 1)= y. The fraction inverts into . Now we can see that y can be any number except 1 so that the range of the original fraction is, indeed, R\{1}.