# What is the domain and range of y=csc x?

Aug 30, 2017

Domain: all $x \in \mathbb{R} : x \ne n \pi \forall n \in \mathbb{Z}$
Range: $\left(- \infty , - 1\right] \cup \left[+ 1 , + \infty\right)$

#### Explanation:

$y = \csc x$

$y = \frac{1}{\sin} x$

$\sin x$ is defined $\forall x \in \mathbb{R}$

$\csc x$ is defined wherever $\sin x \ne 0 \to x \ne n \pi \forall n \in \mathbb{Z}$

Hence, the domain of $y$ is all $x \in \mathbb{R} : x \ne n \pi \forall n \in \mathbb{Z}$

Consider: $- 1 \le \sin x \le + 1 \forall x \in \mathbb{R}$

Hence, the local maxima of $y$ are $\frac{1}{-} 1 = - 1$

and the local minima of $y$ are $\frac{1}{1} = 1$

Then, since $y$ has no upper or lower bounds its range is:
$\left(- \infty , - 1\right] \cup \left[+ 1 , + \infty\right)$

This maybe more easily visualised by the graph of $\csc x$ below.

graph{cscx [-8.89, 8.885, -4.444, 4.44]}