= 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.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2008/paper-1-elementary-graphs.png]
{title=The rational hyperbola and the convex graph of the logarithmic product}
Back to article page
