Question

what are the intervals of y=-2tan2/3x?

Answer 1

I assume this is the function:
\[y=-2\tan \left( \frac{ 2 }{ 3 } x \right) \]

Answer 2

For the tangent function you have to make restrictions. Whenever I work with tangent I just recall that:
\[\tan =\frac{ \sin }{ \cos } \]
This straight away tells me the restriction. As you CANNOT divide by 0. Cos function can take 0s, we do not want that to happen.

Answer 3

Let us just solve this first:
cos x= 0
That is at
\[x=\frac{ \pi }{ 2 }\]
However that is not the full solution as all the trigonometric functions are PERIODIC!
Thus the correct solution is
\[x=\frac{ \pi }{ 2 } +2k \pi \]
where k is any integer,
\[k \in \mathbb{Z} \]

Answer 4

But I solved cosx=0 and for this question we need:
cos(2/3 x)=0

The above helps us greatly:
\[\frac{ 2 }{ 3 }x =\frac{ \pi }{ 2 } +2k \pi \]
Now multiply through by 3/2 to get only x on the left hand side
\[x=\frac{ 3\pi }{ 4 }+3k \pi \]
where \[k \in \mathbb{Z}\]

This is where your function does not make sense, at any other points it is perfectly fine.
The range of tangent is always every number \[R \in \mathbb{R} \]