Question

For what values of x, if any, does `f(x) = tan((3pi)/4-9x)` have vertical asymptotes?

Answer 1

`x=-pi/36` and `x=-pi/18` for `0<=theta<=(2pi)`

Explanation:

Vertical asymptotes occur at values for which the function is undefined. Using this, we know that:

`0<=theta<=(2pi)`

as `theta-> pi, tan(theta)-> oo`

also as `theta-> (3pi)/2, tan(theta)-> -oo`

This means that if:

`((3pi)/4-9x)= pi or (3pi)/2`

The function will be undefined, and vertical asymptotes will occur.

So:

`(3pi)/4-9x= pi=> x=-pi/36`

and:

`(3pi)/4-9x= (3pi)/2=> x=-pi/18`

Graph of `f(x)=tan((3pi)/4-9x)`