Answer
$\color{blue}{(-\infty, 0) \cup (0, +\infty)}$
Work Step by Step
RECALL:
The domain of the sum, difference, and product of $f(x)$ and $g(x)$ is the common elements of the domains of the two functions.
The domain of the respective functions are:
$f(x)$ is defined for all real numbers so its domain is: $(-\infty, +\infty)$
$g(x)$ is defined for all real numbers except $0$ so its domain is: $(-\infty,0) \cup (0 +\infty)$.
Note that:
$[(-\infty, 0) \cup(0, +\infty)] \cap (-\infty, +\infty)
\\= (-\infty, 0) \cup (0, +\infty)$.
Thus, the domain of the sum, difference, and product of $f(x)$ and $g(x)$ is:
$\color{blue}{(-\infty, 0) \cup (0, +\infty)}$
