Determine the domain of the function g:
The function $g$ is defined below.
\[
g(x)=\frac{x-3}{x^{2}-7 x-18}
\]

Step 1 ：The function $g$ is defined as $g(x)=\frac{x-3}{x^{2}-7 x-18}$.

Step 2 ：The domain of a function is the set of all possible input values (x-values) which will output real numbers. In this case, the function is a rational function.

Step 3 ：The denominator of a rational function cannot be zero because division by zero is undefined. Therefore, we need to find the values of x that make the denominator zero and exclude them from the domain.

Step 4 ：The denominator of the function is $x^{2}-7 x-18$. We need to solve the equation $x^{2}-7 x-18 = 0$ to find the values of x that make the denominator zero.

Step 5 ：The solutions to the equation $x^{2}-7 x-18 = 0$ are $x = -2$ and $x = 9$. These are the values that make the denominator of the function zero.

Step 6 ：Therefore, these values should be excluded from the domain of the function.

Step 7 ：\(\boxed{\text{The domain of the function g is all real numbers except -2 and 9. In interval notation, this can be written as } (-\infty, -2) \cup (-2, 9) \cup (9, \infty)}\)