The function provided in the image is:

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

Let me analyze it and provide the solution for its key properties:

Key Steps

1. Factorize the denominator: $x^2 - 3x - 4$ can be written as $(x - 4)(x + 1)$.

2. Simplify the function:

   • Rewrite $f(x)$ as: $f(x) = \frac{(x - 4)^2}{(x - 4)(x + 1)} - 2$
   • For $x \neq 4$: $f(x) = \frac{x - 4}{x + 1} - 2$

3. Combine terms under a common denominator: $f(x) = \frac{x - 4 - 2(x + 1)}{x + 1}$ Simplify the numerator: $f(x) = \frac{x - 4 - 2x - 2}{x + 1} = \frac{-x - 6}{x + 1}, \quad x \neq 4, \, x \neq -1.$

4. Final expression: $f(x) = \frac{-x - 6}{x + 1}, \quad x \neq 4, \, x \neq -1.$

Domain:

• The denominator $x^2 - 3x - 4 = (x - 4)(x + 1) \neq 0$.
• Exclude $x = 4$ and $x = -1$ from the domain.

Thus, the domain is: $x \in \mathbb{R} \setminus \{-1, 4\}.$

Additional Details:

Would you like me to:

1. Analyze vertical and horizontal asymptotes?
2. Plot the function for better visualization?
3. Solve for critical points or intervals of increase/decrease?
4. Find any local maxima/minima?
5. Explore behavior near the excluded points?

Let me know if you have specific questions or need further clarification! One tip: Always simplify rational functions by factoring the numerator and denominator completely.