Question

how to find domain of this function?

Answer 1

The domain of the function, is the \(x\)-values, under which the function is defined.

Answer 2

\(\color{black}{\displaystyle f(x)=\sqrt{\frac{x+1}{x-1}}}\).

It is a square root, and has a denominator.
\(\color{black}{\displaystyle 0\le\frac{x+1}{x-1}}\)
(otherwise, if inside the square root is negative, then \(f\) is undefined)

In order to have
\(\color{black}{\displaystyle 0\le\frac{x+1}{x-1}}\)

either both top and bottom are negative, or both positive.
Thus, you have.
\(\color{black}{\displaystyle x<-1}\) and \(\color{black}{\displaystyle x>1}\)
(Note that this automatically excludes \(x-1=0\))