In order to be in the domain of f(g(x)), x must first be in the domain of g. Then g(x) must be in the domain of f. Here, \(\displaystyle g(x)= \frac{8}{x}+ 3\). Its domain is "all numbers except 0", \(\displaystyle f(x)= \frac{1}{x- 2}\), Its domain is "all numbers except 2". For what x is \(\displaystyle g(x)= \frac{8}{x}+ 3= 2\)?
\(\displaystyle \frac{8}{x}= -1\) so \(\displaystyle x= -8\). The domain of f(g) is "all numbers except 0 and -8". In interval notation that can be written \(\displaystyle (-\infty, -8)\cup(-8, 0)\cup(0, \infty)\).