Find the interval to which a variable in a trigonometric function belongs.

Question

Consider $f\left(x\right)=\tan\left(\frac{3x-\pi }{2}\right)$. Knowing that: $$k\:\in ]0,2\pi] $$ $$\forall x,\:x+k\in \text{Dom}_f,\:f\left(x\right)=f\left(x+k\right)$$

How can we find over which interval $k$ can be found?

Answer 1

To find the period of

$$ f\left(x\right)=\tan\left(\frac{3x-\pi }{2}\right) $$

one must solve the inequality

$$ -\frac{\pi}{2}<\frac{3x-\pi }{2}<\frac{\pi}{2}$$

for $x$.

You will obtain a double inequality of the form

$$ a<x<b$$

and the period $k$ will be found by

$$k=b-a$$

Can you do that? What value do you get for $k$?