the function h is given by h(x) = tan(x + π/6). determine which of the following statements is true. the…

the function h is given by h(x) = tan(x + π/6). determine which of the following statements is true. the locations of the vertical asymptotes of the function h are given by the equation x = π/3 + 2kπ, where k is any integer. the locations of the vertical asymptotes of the function h are given by the equation x = 2π/3 + 2kπ, where k is any integer. the locations of the vertical asymptotes of the function h are given by the equation x = π/3 + kπ, where k is any integer. the locations of the vertical asymptotes of the function h are given by the equation x = 2π/3 + kπ, where k is any integer.

the function h is given by h(x) = tan(x + π/6). determine which of the following statements is true. the locations of the vertical asymptotes of the function h are given by the equation x = π/3 + 2kπ, where k is any integer. the locations of the vertical asymptotes of the function h are given by the equation x = 2π/3 + 2kπ, where k is any integer. the locations of the vertical asymptotes of the function h are given by the equation x = π/3 + kπ, where k is any integer. the locations of the vertical asymptotes of the function h are given by the equation x = 2π/3 + kπ, where k is any integer.

Answer

Explanation:

Step1: Recall tangent - asymptote property

The tangent function $y = \tan(t)$ has vertical asymptotes at $t=\frac{\pi}{2}+k\pi$, where $k\in\mathbb{Z}$.

Step2: Set the argument of the tangent equal to asymptote formula

For $h(x)=\tan(x + \frac{\pi}{6})$, set $x+\frac{\pi}{6}=\frac{\pi}{2}+k\pi$.

Step3: Solve for $x$

Subtract $\frac{\pi}{6}$ from both sides: $x=\frac{\pi}{2}-\frac{\pi}{6}+k\pi$.

Step4: Simplify the right - hand side

$\frac{\pi}{2}-\frac{\pi}{6}=\frac{3\pi - \pi}{6}=\frac{\pi}{3}$, so $x=\frac{\pi}{3}+k\pi,k\in\mathbb{Z}$.

Answer:

The locations of the vertical asymptotes of the function $h$ are given by the equation $x = \frac{\pi}{3}+k\pi$, where $k$ is any integer.