g(x)=cos(3(x+π/6))+2? I think you meant cos not cot, because there is no amplitude associated with cot.
Amplitude=1, period is 2π/3 because the frequency is 3 cycles per period.
However, there are no asymptotes with cos, but there are asymptotes with cot when sine=0, which is when 3(x+π/6)=nπ, that is, when 3x=nπ-π/2, x=nπ/3-π/6=(2n-1)π/6, where n is an integer: -1, 0, 1, 2, so:
x=-π/2, -π/6, π/6, π/2 are asymptotes within the specified interval.
The range for cot is -∞ to +∞, and the domain excludes the above x values where g(x) approaches infinity in both directions. The domain is otherwise [-⅔π,⅔π] for the specified interval.