Question

Given the graph of the function $y=\tan \left(\frac{x}{4}\right),\forall x\in R$
Choose the correct options.

• A. The period of $y$ is $4\pi$ and Domain is $R-n\pi ,n\in Z$
• B. The period of $y$ is $\pi$ and Domain is $R-n\pi ,n\in Z$
• C. The period of $y$ is $\pi$ and Domain is $R-2n\pi ,n\in Z$
• D. The period of $y$ is $4\pi$ and Domain is $R-2n\pi ,n\in Z$

Answer 1

The correct option is D The period of $y$ is $4\pi$ and Domain is $R-2n\pi ,n\in Z$

Given the graph of $y=tan\left(\frac{x}{4}\right)\forall x\in R$

We observe from the graph that the period of $y=4\pi$
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 $\pi$
$\Rightarrow \tan \left(\frac{x}{4}\right)$ will have a fundamental period $\left(\frac{\pi }{1/4}\right)=4\pi$

From the graph we can see that $y$ is not defined at the points where $x$ is an even multiple of $2\pi$, hence
Domain of $y=R-2n\pi ,n\in Z$