# How do you find the domain, range, and asymptote for y = 1 + cot ( 3x + pi/2 )?

Nov 22, 2016

Graph reveals all details. Noe that $\pi = 3.14$, nearly, See explanation, for numerical facts.

#### Explanation:

graph{y-1+tan( 3x)=0 [-10, 10, -5, 5]}

$y = 1 + \cot \left(3 x + \frac{\pi}{2}\right) = 1 - \tan 3 x$

The period for the graph is $\frac{\pi}{3}$. is

y is infinitely discontinuous at x= k/6pi+pi/3), k = 0, +-1, +-2, +-3

The piecewise domain:

$x \in \left(\frac{k}{6} \pi , \frac{k}{6} \pi + \frac{\pi}{3}\right) , k = 0 , \pm 1 , \pm 2 , \pm 3 , . .$.

Range: $y \in \left(- \infty , \infty\right)$.+-3, ...

As $x \to \left(\frac{k}{6} \pi , \frac{k}{6} \pi + \frac{\pi}{3}\right) , k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots y \to \pm \infty$

Asymptotes: $x = \frac{k}{6} \pi + \frac{\pi}{3} , k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$

Have a look at the points inflexion aligned upon y =1,

wherein $x = . \frac{k}{3} \pi , k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$