Question

Given the graph of the function $y=-2\tan \left(3x\right),\forall x\in R$
Choose the correct options.

• A. Period: $\frac{\pi }{3}$
  Domain: $R-\left\{\frac{n\pi }{2}\right\}-\left\{\frac{n\pi }{6}\right\}\forall n\in Z-\left\{0\right\}$
• B. Period: $\frac{\pi }{6}$
  Domain:
  $R-\left\{\frac{n\pi }{2}\right\}-\left\{\frac{n\pi }{6}\right\}\forall n\in Z-\left\{0\right\}$
• C. Period: $\frac{\pi }{3}$
  Domain: $R-\left\{\frac{n\pi }{4}\right\}-\left\{\frac{n\pi }{6}\right\}\forall n\in Z-\left\{0\right\}$
• D. Period: $\frac{\pi }{6}$
  Domain: $R-\left\{\frac{n\pi }{4}\right\}-\left\{\frac{n\pi }{6}\right\}\forall n\in Z-\left\{0\right\}$

Answer 1

The correct option is A Period: $\frac{\pi }{3}$

Domain: $R-\left\{\frac{n\pi }{2}\right\}-\left\{\frac{n\pi }{6}\right\}\forall n\in Z-\left\{0\right\}$
Given the graph of $y=-2\tan \left(3x\right)\forall x\in R$

We observe from the graph that the period of $y=\frac{\pi }{3}$
We also know that if $f\left(x\right)$ has a period $T.$
Then, $f(ax+b)$ will have a period $\frac{T}{|a|}.$

Since we know that $\tan x$ has a fundamental period $\pi$
$\Rightarrow y=-2\tan \left(3x\right)$ will have a fundamental period $\left(\frac{\pi }{3}\right)$

From the graph we can see that $y$ is not defined at the points where $x$ is a multiple of $\frac{\pi }{2}$ and $\frac{\pi }{6}$, hence
Domain of $y$ is $R-\left\{\frac{n\pi }{2}\right\}-\left\{\frac{n\pi }{6}\right\}\forall n\in Z-\left\{0\right\}$