Question

Composite function problem?

Answer 1

If f(x)=\[\sqrt{x+2}\] and (f(g(x))= lxl , find g(x).

Answer 2

Recall that \(\sqrt{x^2}=|x|\), which means for \(f(g(x))=|x|\) to be true, you must have
\[f(g(x))=\sqrt{g(x)+2}~~\iff~~g(x)+2=x^2~~\iff~~g(x)=x^2-2\]
To check, you have
\[f(g(x))=f(x^2-2)=\sqrt{(x^2-2)+2}=\sqrt{x^2}=|x|\]