Graph of the Tangent Function $y = \tan x$

$y = \tan x = \dfrac{\sin x}{\cos x}$ is undefined where $\cos x = 0$, at $x = \dfrac{\pi}{2} + n\pi$, and diverges to $\pm\infty$ near those points (vertical asymptotes).

Its period is $\pi$ and its range is all real numbers. It is an odd function, symmetric about the origin, and near $x = 0$ it behaves like the line of slope $1$.