Question

If f x equals x -3 + g x equals 1 - x 2 - 9 find ( f og) (x and give any restrictions on its domain

Answer 1

$\begin{gathered} f(x)=x-3 \\ g(x)=\frac{1}{x^2-9} \\ (\text{fog)(x)}=\frac{1}{x^2-9}-3 \end{gathered}$

the denominator cannot be zero, because the division by zero is not defined, therefore:

$\begin{gathered} x^2-9=0 \\ \text{Solving for x:} \\ x^2=9 \\ \sqrt[]{x^2}=\sqrt[]{9} \\ x=\pm3 \end{gathered}$

Therefore the domain of (f o g)(x) is:

$D\colon(-\infty,-3)\cup(-3,3)\cup(3,\infty)$