Determine the domain of the function y = tan 2x. Let variable t = 2x, then y = tan 2x changed to y = tan t.

The domain of y = tan t is that t not equal to pi/2 + k pi, in which k = 0, +-1, +-2, so on

t not equal to pi/2 + k pi

substitute t = 2x

2x not equal to pi/2 + k pi

divide by 2 in each term

x not equal to pi/4 + k pi/2

when k = -1, x not equal to pi/4 + (-1) × pi/2 -> x not equal to - pi/4

when k = 0, x not equal to pi/4 + 0 × pi/2 -> x not equal to pi/4

when k = 1, x not equal to pi/4 + 1 × pi/2 -> x not equal to 3pi/4

when k = 2, x not equal to pi/4 + 2 × pi/2 -> x not equal to 5pi/4

The asymptotes of y = tan 2x is

when k = -1, x = -pi/4

when k = 0, x = pi/4

when k = 1, x = 3pi/4

when k = 2, x = 5pi/4

In the figure above, the vertical lines are the asymptotes of the function y = tan 2x. The blue curve is the graph of y = tan 2x. The graph of y = tan 2x is never across its asymptotes.
The minimum period of the function y = tan 2x is pi/2.

In the figure above, the blue vertical lines are the asymptotes of the function y = tan 2x. The blue curve is the graph of the function y = tan 2x. The blue curve is never across the blue asymptotes.

The orange vertical lines are asymptotes of the function y = tan x. The orange curve is the graph of y = tan x. The orange curve is never across the orange asymptotes.

The asymptotes of y = tan 2x is the asymptotes of y = tan x shrink by half along the x-axis. The graph of y = tan 2x is the graph of y = tan x shrinks by half along the x-axis.