(tangent function behavior lc) given the function g(x) = tan x, determine the location of the asymptote on…

Answer:

$x=\frac{9\pi}{2}$

Explanation:

Step1: Recall tangent - asymptote formula

The asymptotes of $y = \tan x$ occur at $x=(n+\frac{1}{2})\pi$, where $n$ is an integer.

Step2: Find $n$ for the given interval

We want to find $n$ such that $4\pi<(n + \frac{1}{2})\pi<5\pi$. First, divide the entire inequality by $\pi$: $4 < n+\frac{1}{2}<5$. Then subtract $\frac{1}{2}$ from all parts: $4-\frac{1}{2}<n<5 - \frac{1}{2}$, so $\frac{7}{2}<n<\frac{9}{2}$. Since $n$ is an integer, $n = 4$.

Step3: Calculate the asymptote

Substitute $n = 4$ into the formula $x=(n+\frac{1}{2})\pi$. We get $x=(4+\frac{1}{2})\pi=\frac{9\pi}{2}$.