Question

Find domain of tanx/tan3x

Answer 1

tanx/tan3x
=I1−3tan^2x)/(3−tan^2x)_
=3(3−tan^2x)−8)/(3−tan^2x)
=(1-3tan^2x)/(3−tan^2x)

Now let z = tanx
f(z)=(1-3z^2/(3-z^2)

so it looks like z cannot be sqrt(3), or
tan x cannot be sqrt3, or x cannot be....

Answer 2

the function is undefined where the denominator is zero. That is,

tan(3x) = 0
this occurs when 3x = nπ
or, x = n * π/3

where x=nπ the function is 0/0, so there are holes there.

the function is also undefined when the numerator or denominator is undefined. That is, when

3x or x is an odd multiple of π/2

If you look at the graph, all you see are the asymptotes, but the holes are also excluded from the domain.