$f(x) = \frac{{\sqrt 2 x}}{{3x - \sqrt 2 }} \to f(\sqrt 2 ) = \frac{2}{{3\sqrt 2 - \sqrt 2 }} = \frac{1}{{\sqrt 2 }}$
$ \to f(f(\sqrt 2 )) = f\left( {\frac{1}{{\sqrt 2 }}} \right) = \frac{1}{{\frac{3}{2} - \sqrt 2 }} = \frac{{\sqrt 2 }}{{3 - 2}} = \sqrt 2 $
$ \to f(\underbrace {f(f(\sqrt 2 ))}_{\sqrt 2 }) = f(\sqrt 2 ) = \frac{1}{{\sqrt 2 }}$