No Dividing by Zero

A rule with a fraction loses every input that makes its denominator zero. To find them, set the denominator equal to zero and solve. For $f\left(x\right) = \frac{5}{x-3}$, the equation $x-3 = 0$ gives 3, so 3 is the one input f refuses.

The numerator does not matter here. Zero on top is fine: $\frac{0}{4}$ is just 0. Only zero underneath breaks the rule.

For $g\left(x\right) = \frac{1}{3x-6}$, solve $3x-6 = 0$: 3x is 6, so x is 2. The domain of g is every real number except 2.