Study of a homographic function

Text

Study the homographic function $y=\dfrac{x-1}{x+2}$: domain, asymptotes, axis intercepts and monotonicity.

Solution

Domain: $x\neq-2$. Vertical asymptote: $x=-2$; horizontal asymptote: $\displaystyle\lim_{x\to\pm\infty}\dfrac{x-1}{x+2}=1$, so $y=1$. Intercepts: $x$-axis at $(1,0)$; $y$-axis at $\big(0,-\tfrac12\big)$. First derivative: $y'=\dfrac{3}{(x+2)^{2}}>0$: always increasing on each branch, no extrema. The graph is a rectangular hyperbola with centre $(-2,1)$.