Answer
No.
Work Step by Step
Use the definition of inverses, for $f(x)=\frac{-1}{x+1}$ and $g(x)=\frac{1-x}{x}$, we have:
$f(g(x))=\frac{-1}{\frac{1-x}{x}+1}=\frac{-x}{1-x+x}=-x\ne x$
Thus f and g are not inverses to each other.
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