Determine the domain of the functions f and g:
The functions $f$ and $g$ are defined as follows.
\[
\begin{array}{l}
f(x)=\frac{x-3}{x^{2}-10 x+21} \\
g(x)=\frac{x^{2}}{x-9}
\end{array}
\]

Step 1 ：The functions $f$ and $g$ are defined as follows: $f(x)=\frac{x-3}{x^{2}-10 x+21}$ and $g(x)=\frac{x^{2}}{x-9}$.

Step 2 ：The domain of a function is the set of all possible input values (often the 'x' variable), which produce a valid output from a particular function.

Step 3 ：For these functions, we need to find the values of x for which the functions are defined.

Step 4 ：For the function $f(x)$, the denominator cannot be equal to zero because division by zero is undefined in mathematics. Therefore, we need to find the values of x that make the denominator zero and exclude them from the domain.

Step 5 ：For the function $g(x)$, the denominator also cannot be zero. So, we need to find the value of x that makes the denominator zero and exclude it from the domain.

Step 6 ：The solutions for the denominators of the functions $f(x)$ and $g(x)$ are $x = 3, 7$ and $x = 9$ respectively. These are the values that make the denominators zero and hence, they should be excluded from the domain of the functions.

Step 7 ：Therefore, the domain of the function $f(x)$ is all real numbers except 3 and 7. The domain of the function $g(x)$ is all real numbers except 9.

Step 8 ：Final Answer: The domain of the function $f(x)$ is \(\boxed{\mathbb{R} \setminus \{3, 7\}}\) and the domain of the function $g(x)$ is \(\boxed{\mathbb{R} \setminus \{9\}}\).