which is an asymptote of the graph of the function $y = \tan(\frac{3}{4}x)$?\n$x=-\frac{4pi}{3}$\n$x =…

Explanation:

Step1: Recall tangent - asymptote formula

The asymptotes of the tangent function $y = \tan(u)$ occur when $u=(n +\frac{1}{2})\pi$, where $n\in\mathbb{Z}$. For the function $y=\tan(\frac{3}{4}x)$, we set $\frac{3}{4}x=(n+\frac{1}{2})\pi$.

Step2: Solve for $x$

Multiply both sides of the equation $\frac{3}{4}x=(n+\frac{1}{2})\pi$ by $\frac{4}{3}$. So $x=\frac{4}{3}(n +\frac{1}{2})\pi=\frac{4n\pi}{3}+\frac{2\pi}{3}$. When $n=- 1$, $x=\frac{-4\pi}{3}+\frac{2\pi}{3}=-\frac{2\pi}{3}$. When $n = - 2$, $x=\frac{-8\pi}{3}+\frac{2\pi}{3}=-2\pi$. When $n = 0$, $x=\frac{2\pi}{3}$. When $n=-1$ in the general form of the asymptote $x=\frac{4}{3}(n+\frac{1}{2})\pi$, we get $x =-\frac{2\pi}{3}$.

Answer:

$x =-\frac{2\pi}{3}$