Answer
No.
Work Step by Step
The two functions are inverses if:
$(f\circ g)(x)=x$ and $(g\circ f)(x) = x$;
$x\ne (-6) x\ne 0$
Here,
$f(g(x))=\frac{2}{\frac{6x+2}{x}+6}$.
After multiplying $x$ to both the numerator and the denominator, we get:
$\frac{2x}{6x+2+6x}=\frac{2x}{12x+2}\ne x$
Therefore, $f$ and $g$ are not inverses.