Answer:
The correct option is D.
i.e.
[tex]f\left(x\right)=\frac{1}{x\left(x+4\right)}[/tex] is the correct option.
The correct graph is shown in attached figure.
Step-by-step explanation:
Considering the function
[tex]f\left(x\right)=\frac{1}{x\left(x+4\right)}[/tex]
[tex]\mathrm{Domain\:of\:}\:\frac{1}{x\left(x+4\right)}\::\quad \begin{bmatrix}\mathrm{Solution:}\:&\:x<-4\quad \mathrm{or}\quad \:-4<x<0\quad \mathrm{or}\quad \:x>0\:\\ \:\mathrm{Interval\:Notation:}&\:\left(-\infty \:,\:-4\right)\cup \left(-4,\:0\right)\cup \left(0,\:\infty \:\right)\end{bmatrix}[/tex]
[tex]\mathrm{Range\:of\:}\frac{1}{x\left(x+4\right)}:\quad \begin{bmatrix}\mathrm{Solution:}\:&\:f\left(x\right)\le \:-\frac{1}{4}\quad \mathrm{or}\quad \:f\left(x\right)>0\:\\ \:\mathrm{Interval\:Notation:}&\:(-\infty \:,\:-\frac{1}{4}]\cup \left(0,\:\infty \:\right)\end{bmatrix}[/tex]
[tex]\mathrm{Axis\:interception\:points\:of}\:\frac{1}{x\left(x+4\right)}:\quad \mathrm{None}[/tex]
[tex]\mathrm{Extreme\:Points\:of}\:\frac{1}{x\left(x+4\right)}:\quad \mathrm{Maximum}\left(-2,\:-\frac{1}{4}\right)[/tex]
So, the correct graph is shown in attached figure.
Therefore, the correct option is D.
i.e.
[tex]f\left(x\right)=\frac{1}{x\left(x+4\right)}[/tex] is the correct option.
