Answer
No.
Work Step by Step
Use the definition of inverses, for $f(x)=\frac{2}{x+6}$ and $g(x)=\frac{6x+2}{x}$, we have:
$f(g(x))=\frac{2}{\frac{6x+2}{x}+6}=\frac{2x}{6x+2+6x}=\frac{2x}{12x+2}\ne x$
Thus f and g are not inverses to each other.