Question

For what values of x, if any, does `f(x) = -tan(pi/6-x)` have vertical asymptotes?

Answer 1

`f(x)` have Vertical Asymptotes for `x=-pi/3+-n*180`

where `n=0, 1, 2, 3.....`

Explanation:

From the given `f(x)=-tan(pi/6-x)`

Set the angle `(pi/6-x)=pi/2` because `tan (pi/2)=`undefined

Solving for `x`

`x=-pi/3`

and for tangent function the values are the same everytime `pi` or multiples of `pi` are added or subtracted.
Therefore, the function has asymptotes for values of
`x=pi/3+-n*pi` where `n=0, 1, 2, 3,.....`
Examples:

`x=-pi/3`

`x=-(4pi)/3`

`x=(2pi)/3`

`x=(5pi)/3`