identify the period and equation of one asymptote of the function f(x)=3 csc(1/3 x)-2. period: equation of…

Explanation:

Step1: Recall csc - function period formula

The period of $y = A\csc(Bx - C)+D$ is $\frac{2\pi}{|B|}$. For $f(x)=3\csc(\frac{1}{3}x)-2$, $B = \frac{1}{3}$.

Step2: Calculate the period

Substitute $B=\frac{1}{3}$ into the period - formula: $\frac{2\pi}{\left|\frac{1}{3}\right|}=6\pi$.

Step3: Recall csc - function asymptote formula

The asymptotes of $y = \csc(x)$ occur at $x = n\pi$, where $n\in\mathbb{Z}$. For $y=\csc(Bx - C)+D$, the asymptotes are given by $Bx - C=n\pi$. In $f(x)=3\csc(\frac{1}{3}x)-2$, $B=\frac{1}{3}$ and $C = 0$. Setting $\frac{1}{3}x=n\pi$, when $n = 0$, one asymptote is $x = 0$.

Answer:

Period: $6\pi$ Equation of one asymptote: $x = 0$