 Solution

Write $x/(x-1)=1+1/(x-1)$. The graph is a translated hyperbola, with vertical asymptote $x=1$ and horizontal asymptote $y=1$. It passes through $(0,0)$, lies below $y=1$ for $x<1$ and above it for $x>1$, and has derivative $-1/(x-1{)}^2<0$ on each component of its domain. The left panel below shows both branches.