Answer
Since $f\circ g(x)=x$ and $g\circ f(x)=x $
f and g are inverses of each other.
Work Step by Step
$\begin{array}{lllll}
f\circ g(x) & =f[g(x)] & ..... & g\circ f(x) & =g[f(x)]\\\\
& =\dfrac{1}{g(x)-1} & & & =\dfrac{1}{f(x)}+1\\\\
& =\dfrac{1}{\frac{1}{x}+1-1} & & & =\dfrac{1}{\frac{1}{x-1}}+1\\\\
& =\dfrac{1}{\frac{1}{x}} & & & =(x-1)+1\\
& =x & & & =x
\end{array}$
Since $f\circ g(x)=x$ and $g\circ f(x)=x $
f and g are inverses of each other.