Question

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

Answer 1

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

We observe from the graph that the period of $y\text{is}4\pi$, as the graph repeats itself after every $4\pi$ interval.


From the graph we can see that $y$ is not defined at the points where $x$ is a multiple of $2(2n+1)\pi$,
We also know that the domain of $\tan \theta$ is $R-\frac{(2n+1)\pi }{2}$, therefore the domain of $\tan \left(\frac{x}{4}\right)is4(R-\frac{(2n+1)\pi }{2}),\text{i.e.},R-2(2n+1)\pi$
hence
Domain of $y=R-2(2n+1)\pi$