The function y = tan(3x/4) has vertical asymptotes where the argument of the tangent function, 3x/4, is equal to (2n + 1)π/2, where n is an integer. This happens because the tangent function approaches infinity at these points.
To find the asymptotes, we can set up the equation:
3x/4 = (2n + 1)π/2
Solving for x, we multiply both sides by 4/3:
x = (4/3)((2n + 1)π/2) = (2(2n + 1)π)/3
Thus, the vertical asymptotes are located at:
x = (2(2n + 1)π)/3, for n = 0, ±1, ±2, …
For example, when n = 0, the asymptote is at x = (2π)/3. When n = 1, it is at x = (6π)/3 which simplifies to 2π, and so on. You can find an infinite number of these vertical asymptotes along the x-axis.