How do you find the domain & range for  3 Cot(2x)?

Jul 23, 2018

Domain : $x \in \left(- \infty , \infty\right)$, sans $\uparrow$ asymptoiic $\downarrow$ $x = k \left(\frac{\pi}{2}\right)$,
$k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$
Range: $y \notin \left(- \frac{1}{2} , \frac{1}{2}\right)$

Explanation:

$y = 3 \cot 2 x = 3 \left(\cos \frac{2 x}{\sin} \left(2 x\right)\right) \in \left(- \infty . \infty\right)$, ,

$x \ne$ zeros of the denominator $\sin 2 x$

$\Rightarrow x \ne$ an integer multiple of $\left(\frac{\pi}{2}\right)$

The period

= the common period of $\sin 2 x \mathmr{and} \cos 2 x = \frac{2 \pi}{2} = \pi$

See graph. revealing all these aspects:
graph{(y sin (2x) - 3 cos (2x) )= 0[ -10 10 -10 10]}