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

which is an asymptote of the graph of the function $y = \\tan(\\frac{3}{4}x)$?\n$x = -\\frac{4\\pi}{3}$\n$x = -\\frac{2\\pi}{3}$\n$x = \\frac{3\\pi}{4}$\n$x = \\frac{3\\pi}{2}$

which is an asymptote of the graph of the function $y = \\tan(\\frac{3}{4}x)$?\n$x = -\\frac{4\\pi}{3}$\n$x = -\\frac{2\\pi}{3}$\n$x = \\frac{3\\pi}{4}$\n$x = \\frac{3\\pi}{2}$

Answer

Answer:

A. $x =-\frac{4\pi}{3}$

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 $\frac{4}{3}$ to get $x=\frac{4}{3}(n +\frac{1}{2})\pi=\frac{4n\pi}{3}+\frac{2\pi}{3}$.

Step3: Find a specific asymptote

When $n=- 2$, $x=\frac{4\times(-2)\pi}{3}+\frac{2\pi}{3}=\frac{-8\pi + 2\pi}{3}=-\frac{4\pi}{3}$.