Question

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

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

Answer 1

The correct option is C Period of $y:\frac{\pi }{2}$ and Domain of $y:R-\left\{\frac{n\pi }{2}\right\}\forall n\in Z-\left\{0\right\}$

Given the graph of $y=3\cot \left(2x\right)\forall x\in R$
We observe from the graph that the period of $y=\frac{\pi }{2}$
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 $\cot x$ has a fundamental period $\pi$
$\Rightarrow y=3\cot \left(2x\right)$ will have a fundamental period $\left(\frac{\pi }{2}\right)$

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