Explanation
Part A
We are told to obtain the domain of the given function:
[tex]w(x)=|x+1|+4[/tex]
The domain of a function is the set of all possible inputs for the function.
we can use the graph to obtain the intervals of the domain
[tex]\mathrm{The\:domain\:of\:a\:function\:is\:the\:set\:of\:input\:or\:argument\:values\:for\:which\:the\:function\:is\:real\:and\:defined}[/tex]
Thus, the solution is
[tex]\begin{bmatrix}\mathrm{Solution:}\:&\:-\infty \:
For part B
[tex]y(x)=\frac{x}{|x+1|+4}[/tex]
The domain will be
[tex]\begin{bmatrix}\mathrm{Solution:}\:&\:-\infty \:
Part C
[tex]z(x)=\frac{x}{|x+1|-4}[/tex]
The domain will be
[tex]\begin{bmatrix}\mathrm{Solution:}\:&\:x<-5\quad \mathrm{or}\quad \:-53\:\\ \:\mathrm{Interval\:Notation:}&\:\left(-\infty \:,\:-5\right)\cup \left(-5,\:3\right)\cup \left(3,\:\infty \:\right)\end{bmatrix}[/tex]
