To find the domain of each function $f(x)$ and $g(x)$, we need to determine the values of $x$ that would make the denominator zero, as division by zero is undefined.

For $f(x) = \frac{x - 3}{x^2 - 7x + 12}$

1. Factor the denominator: $x^2 - 7x + 12 = (x - 3)(x - 4)$ So, $f(x) = \frac{x - 3}{(x - 3)(x - 4)}$.

2. Identify values that make the denominator zero:

   • $x = 3$ and $x = 4$ both make the denominator zero.

3. Exclude these values from the domain.

Thus, the domain of $f(x)$ is: $(-\infty, 3) \cup (3, 4) \cup (4, \infty)$

For $g(x) = \frac{x^2}{x - 2}$

1. Identify values that make the denominator zero:

   • $x = 2$ makes the denominator zero.

2. Exclude $x = 2$ from the domain.

Thus, the domain of $g(x)$ is: $(-\infty, 2) \cup (2, \infty)$

Summary of Domains:

• Domain of $f(x)$: $(-\infty, 3) \cup (3, 4) \cup (4, \infty)$
• Domain of $g(x)$: $(-\infty, 2) \cup (2, \infty)$

Would you like more details on these steps, or have any questions?

Here are 5 related questions to deepen understanding:

1. What happens to $f(x)$ near $x = 3$ and $x = 4$?
2. Can we simplify $f(x)$ if $x \neq 3$?
3. How would you find the range of $f(x)$ and $g(x)$?
4. Why do rational functions have domain restrictions at zero denominators?
5. What would the graph of $g(x)$ look like near $x = 2$?

Tip: When finding domains, always start by identifying any values that make the denominator zero for rational functions.