Answer:
he domain of the composition is all real x values except for x = -1
In other words: [tex]\left \{ x \, |\, x \neq -1} \right \}[/tex]
Step-by-step explanation:
Let's find the composition [tex]f(g(x))[/tex] in order to answer about its domain (where on the Real number set the function is defined), give the two functions:
[tex]f(x)= \frac{1}{x+4}[/tex] and [tex]g(x)=\frac{8}{x-1}[/tex] :
[tex]f(g(x))=\frac{1}{g(x)+4} \\f(g(x))=\frac{1}{\frac{8}{x-1} +4} \\f(g(x))=\frac{1}{\frac{8+4(x-1)}{x-1} }\\f(g(x))=\frac{x-1}{8+4x-4} \\f(g(x))=\frac{x-1}{4+4x} \\[/tex]
This rational function is defined for every real number except when the denominator adopts the value zero. Such happens when:
[tex]4+4x=0\\4x=-4\\x=-1[/tex]
So the domain of the composition is all real x values except for x = -1
