For l(x) to make sense, it is required that x−1x−3∈(−1,1).
Then, we solve the inequality −1<x−1x−3<1 to find the domain of f.
I will do one side of the inequality as an example. Let's say we want to solve x−1x−3<1. We can't just multiply (x - 3) on both sides because it is not guaranteed that x - 3 is positive. Instead, we multiply (x - 3)^2 on both sides since it is always nonnegative. Also, note that x cannot be 3 because of the denominator.
x−1x−3⋅(x−3)2≤(x−3)2,x≠3(x−1)(x−3)≤(x−3)2,x≠3x2−4x+3≤x2−6x+9,x≠32x−6≤0,x≠32x−6<0x<3
Now, you can solve the other inequality −1<x−1x−3 to get another inequality for x, combine them, and you get the domain of l(x).