Explanation:
For the given function
[tex]f(x)=\frac{1}{(x+3)}-2[/tex]
We are told to Graph the function
The graph is given below
Thus
The Domain of the function is
[tex]\begin{bmatrix}\mathrm{Solution\colon}\: & \: x<-3\quad \mathrm{or}\quad \: x>-3\: \\ \: \mathrm{Interval\: Notation\colon} & \: \mleft(-\infty\: ,\: -3\mright)\cup\mleft(-3,\: \infty\: \mright)\end{bmatrix}[/tex]
The Range of the function is
[tex]\begin{bmatrix}\mathrm{Solution\colon}\: & \: f\mleft(x\mright)<-2\quad \mathrm{or}\quad \: f\mleft(x\mright)>-2\: \\ \: \mathrm{Interval\: Notation\colon} & \: \mleft(-\infty\: ,\: -2\mright)\cup\mleft(-2,\: \infty\: \mright)\end{bmatrix}[/tex]
The Asymptotes are
[tex]\begin{gathered} \mathrm{Vertical}\colon\: x=-3,\: \\ \mathrm{Horizontal}\colon\: y=-2 \end{gathered}[/tex]
It is decreasing on
[tex]\mathrm{Decreasing}\colon-\infty\:
It is increasing on
[tex]\text{None}[/tex]
Limits
[tex]\begin{gathered} \lim _{x\Rightarrow3^-}f(x)=-\infty \\ \\ \lim _{x\Rightarrow3^+}f(x)=\infty \end{gathered}[/tex]
