Question

Given f(x) = 1/x+4 and g(x) = 8/x-1, find the given domain of f(g(x)).

Answer 1

Answer:

he domain of the composition is all real x values except for x = -1

In other words: $\left \{ x \, |\, x \neq -1} \right \}$

Step-by-step explanation:

Let's find the composition $f(g(x))$ in order to answer about its domain (where on the Real number set the function is defined), give the two functions:

$f(x)= \frac{1}{x+4}$    and    $g(x)=\frac{8}{x-1}$  :

$f(g(x))=\frac{1}{g(x)+4} \\f(g(x))=\frac{1}{\frac{8}{x-1} +4} \\f(g(x))=\frac{1}{\frac{8+4(x-1)}{x-1} }\\f(g(x))=\frac{x-1}{8+4x-4} \\f(g(x))=\frac{x-1}{4+4x}$

This rational function is defined for every real number except when the denominator adopts the value zero. Such happens when:

$4+4x=0\\4x=-4\\x=-1$

So the domain of the composition is all real x values except for x = -1