graph the following and state the range and all vertical asymptotes in equation form. show work…

Explanation:

Step1: Identify the general form parameters

The general form of the cotangent - function is $y = a\cot(b(x - c))+d$. For the function $f(x)=\cot(\frac{1}{2}(x + \pi))+3$, we have $a = 1$, $b=\frac{1}{2}$, $c=-\pi$, $d = 3$.

Step2: Find the range

The range of the basic cotangent function $y=\cot(x)$ is $(-\infty,\infty)$. A vertical shift of $d = 3$ units does not change the range. So the range of $y=\cot(\frac{1}{2}(x+\pi)) + 3$ is $(-\infty,\infty)$.

Step3: Find the vertical asymptotes

The vertical asymptotes of the basic cotangent function $y = \cot(x)$ occur at $x=n\pi$, where $n\in\mathbb{Z}$. For the function $y=\cot(b(x - c))+d$, the vertical - asymptotes are given by the equation $b(x - c)=n\pi$. Substitute $b=\frac{1}{2}$ and $c = -\pi$ into $b(x - c)=n\pi$: [ \begin{align*} \frac{1}{2}(x+\pi)&=n\pi\ x+\pi&=2n\pi\ x&=2n\pi-\pi=(2n - 1)\pi,n\in\mathbb{Z} \end{align*} ]

Answer:

$a = 1$, $b=\frac{1}{2}$, $c=-\pi$, $d = 3$ Range: $(-\infty,\infty)$ Vertical Asymptotes: $x=(2n - 1)\pi,n\in\mathbb{Z}$