Question

For what values of x, if any, does `f(x) = tan((-7pi)/4-2x)` have vertical asymptotes?

Answer 1

`x = ( 2 k + 1 ) pi/4 - 7/8 pi = (4 k - 5 ) pi/8,`
`k = 0, +-1, +-2, +-3, ...`
`x = ... , - 5/8 pi, - 1/8 pi, 3/8 pi, 7/8 pi,...`

Explanation:

`y = tan ( -7/4 pi - 2 x) = - tan ( 2 ( x + 7/8 pi )`

The period = `pi/2`.
The vertical asymptotes of `y = a tan (bx + c) + d` are

`b x + c = ( 2 k + 1 ) pi/2, k = 0, +-1, +-2, +-3, ...`

Here, they are given by

`2 ( x + 7/8 pi ) = ( 2 k + 1 ) pi/2`

`rArr x = ( 2 k + 1 ) pi/4 - 7/8 pi = (4 k - 5 ) pi/8,`

`k = 0, +-1, +-2, +-3, ...`

`= ..., , - 5/8 pi, - 1/8 pi, 3/8 pi...`

See graph:
graph{(y+tan(2(x+7/8pi)))=0[-pi pi -pi/2 pi/2]}