#### Answer

$f$ and $g$ are not inverses of each other.

#### Work Step by Step

RECALL:
(1) $(f \circ g)(x) = f\left(g(x)\right)$
(2) The function $g(x)$ is the inverse of function of a one-to-one function $(x)$ if for every element of the domain,
$(f \circ g)(x) =x$ and $(g \circ f)(x)=x$
Find $(f \circ g)(x)$ by substituting $\frac{2x+1}{x-1}$ to $x$ in $f(x)$:
$(f\circ g)(x)
\\=f\left(g(x)\right)
\\=\dfrac{2x+1+1}{2x+1-2}
\\=\dfrac{2x+2}{2x-1}$
Since $(f\circ g)(x)\ne x$, then $f$ and $g$ are not inverses of each other.