The function you've provided is:

$f(x) = \frac{x^2 + 3x - 4}{x^2 - 7x + 12}$

Step 1: Factor the numerator and denominator

First, let's factor both the numerator and the denominator, if possible.

Numerator: $x^2 + 3x - 4$

We need to factor this quadratic. We are looking for two numbers that multiply to $-4$ (the constant term) and add to $3$ (the coefficient of $x$).

The factors of $-4$ that sum to $3$ are $4$ and $-1$. Thus, the numerator factors as:

$x^2 + 3x - 4 = (x + 4)(x - 1)$

Denominator: $x^2 - 7x + 12$

Now, we factor the denominator. We are looking for two numbers that multiply to $12$ (the constant term) and add to $-7$ (the coefficient of $x$).

The factors of $12$ that sum to $-7$ are $-3$ and $-4$. Thus, the denominator factors as:

$x^2 - 7x + 12 = (x - 3)(x - 4)$

Step 2: Simplify the function

Now, the function becomes:

$f(x) = \frac{(x + 4)(x - 1)}{(x - 3)(x - 4)}$

Notice that the factor $(x - 4)$ appears in both the numerator and the denominator, so it can be canceled, but we must be careful to note that $x \neq 4$, because division by zero is undefined.

Thus, the simplified function is:

$f(x) = \frac{x + 4}{x - 3}, \quad \text{for} \, x \neq 4$

Step 3: Domain of the function

The function is undefined when the denominator is zero. Setting the original denominator equal to zero:

$x^2 - 7x + 12 = 0$ $(x - 3)(x - 4) = 0$

So, $x = 3$ and $x = 4$ are the points where the function is undefined. Therefore, the domain of the function is:

$x \in \mathbb{R}, \quad x \neq 3, 4$

Final Answer:

$f(x) = \frac{x + 4}{x - 3}, \quad x \neq 3, 4$

Would you like to explore further details or any specific steps in this process?

Here are 5 related questions:

1. How would you find vertical asymptotes for this function?
2. What are the horizontal asymptotes of rational functions?
3. Can we determine the end behavior of $f(x)$?
4. How would you graph this function?
5. How can we determine the range of $f(x)$?

Tip: When simplifying rational functions, always check for values that make the denominator zero and exclude them from the domain.