Graph f (x)=tan 3x. identify period, x-intercepts and asymptotes

Well the period of the tan(x) function is π. It means that when you add π on X axis you will get the same y axis value.

There is a little trick to find the period of functions like yours.
You just divide period of basic tan(x)  function with, in your case 3.
So this mean you have
To = π/3

Function intercepts x axis when function is equal to 0.

tan(3x) = 0
3x = k * π
x = k * (π / 3)

There are only vertical asymptotes in tangent function.
You can easily see it if you look at the graph

 tan(x) = ∞ ,          (π/2 + kπ)
 tan(x) = -∞           (-π/2 + kπ).
x = π/2 + kπ
x = - π/2 + kπ

tan(3x)  = ∞

3x = π/2 + kπ
x = π/6 + kπ/3

tan(3x)  = -∞

3x = -π/2 + kπ
x = -π/6 + kπ/3