Question

FIND f(g(x)) and g(f(x)) and each of the domains?

f(x)=sqrt(x+4) , g(x)=x^2 -4

Answer 1

Do you know how to start?

Answer 2

I know how to solve them, but I mostly need help with finding the domains. Would you substitute g(x) for x in f(x) and vice versa? Both times getting x after simplifying?

Answer 3

\[f(x)=\sqrt{x+4}~~~~~~~~~g(x)=x^2-4\]
So
\[f(g(x))=\sqrt{(x^2-4)+4}=\sqrt{x^2}=|x|,~~\text{not }x\]
and
\[g(f(x))=\left(\sqrt{x+4}\right)^2-4=x+4-4=x\]

Answer 4

Right, and the x in f(g(x)) would be in between the absolute values, because it was in a square root?

Answer 5

Yeah. It has to do with the sign on the inside of the square root. Can't have \(x^2<0\).

But the domains of the composite functions don't depend on \(x\) or \(|x|\), but the domains of the component functions.