determine amplitude, period, phase shift, vertical shift, asymptotes, domain & range

Question

for the function

g(x)=cot3(x+pi/6)+2 ; [-2pi/3, 2pi/3]

Answer 1

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.