Let us write our functions:
[tex]f(x)=x^2-4,g(x)=\sqrt[]{x-7}[/tex]Now,
[tex](f○g)(x)=f(g(x))=f(\sqrt[]{x-7})=(\sqrt[]{x-7})^2-4=(x-7)-4=x-11[/tex]Shorter,
[tex](f○g)(x)=x-11[/tex]On the other hand, we need to remember that the domain of a composition is the domain of the first function to be composed; in this case, it's g(x). The domain of g(x) consists of all numbers x such that
[tex]x-7\ge0[/tex]for the square root of negative numbers makes no sense. Then,
[tex]x\ge7[/tex]The domain of g(x) consists of all numbers greater than or equal to 7, and the domain of f(g(x)) as well, by what we said above.
