Question

Use the pair of functions to find f(g(x)) and g(f(x)). Simplify your answers. F(X)=sqrt x+8, g(x)=x^2+3

Answer 1

Recall that:

$\begin{gathered} f(g(x))=f(x^2+3), \\ g(f(x))=g(\sqrt{x}+8). \end{gathered}$

Evaluating the functions, we get:

$\begin{gathered} f(g(x))=\sqrt{x^2+3}+8, \\ g(f(x))=(\sqrt{x}+8)^2+3. \end{gathered}$

Simplifying the second function, we get:

$g(f(x))=x+16\sqrt{x}+64+3=x+16\sqrt{x}+67.$

Answer:

$\begin{gathered} f(g(x))=\sqrt{x^2+3}+8, \\ g(f(x))=x+16\sqrt{x}+67. \end{gathered}$