"f+ g" is defined by "f+ g(x)= f(x)+ g(x)". In order that f+ g (x) exist, f(x) and g(x) must both exist (even if adding algebraically appears to eliminate a "division by 0").
That means that the domain of f+ g is the intersection of the domains of f and g. The domain of f(x)= 8/(x- 9) (which is NOT what you wrote but must be what you meant since the domain of 8/x- 9 is "all real numbers except 0") is "all real numbers except 9" and the domain of g(x)= 14/(x+ 3) is "all real numbers except -3". The domain of f+ g is "all real numbers except 9 and -3". In interval notation, that would be .
For f/g, both f and g must exist and g(x) must not be 0. Since 14/(x+ 3) is never 0 (a fraction is equal to 0 if and only if its numerator is 0), the domain of f/g is exactly the same as f+ g.