Questions: Sketch the graph of (y=csc x), and then identify as many of the following properties that apply. Answer (a) through (e).
(a) Is the function an odd function?
yes
no
(b) Does the function obtain a relative maximum at (x=pi n) where n is an odd integer?
yes
no
(c) Is the domain all real numbers except integer multiples of (pi)?
no
yes
(d) Are the equations of the vertical asymptotes of the form (x=n pi) where (n) is an integer?
no
yes
(e) Does the function have a period of (P=pi)?
yes
Transcript text: Sketch the graph of $y=\boldsymbol{\operatorname { c s c }} x$, and then identify as many of the following properties that apply. Answer (a) through (e).
(a) Is the function an odd function?
yes
no
(b) Does the function obtain a relative maximum at $\mathrm{x}=\pi \mathrm{n}$ where n is an odd integer?
yes
no
(c) Is the domain all real numbers except integer multiples of $\pi$ ?
no
yes
(d) Are the equations of the vertical asymptotes of the form $x=n \pi$ where $n$ is an integer?
no
yes
(e) Does the function have a period of $P=\pi$ ?
yes
Solution
Solution Steps
Step 1: Determine if the function is odd
The function \( y = \csc x \) is an odd function because \(\csc(-x) = -\csc(x)\).
Step 2: Check for relative maximum at \( x = \pi n \)
The function \( y = \csc x \) does not obtain a relative maximum at \( x = \pi n \) where \( n \) is an odd integer. At these points, the function is undefined.
Step 3: Determine the domain of the function
The domain of \( y = \csc x \) is all real numbers except integer multiples of \(\pi\), because \(\csc x = \frac{1}{\sin x}\) and \(\sin x = 0\) at integer multiples of \(\pi\).