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

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Answer 1 · 2018-08-02T14:41:02

# 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,...#

#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]}