8. \( y=\frac{x-3}{x^{2}-7 x+12} \) (a) \( \{x \in \mathbf{R} \mid x=3, x \neq 4\} \) (b) \( \{x \in \mathbf{R} \mid x \neq 3\} \) (c) \( \{x \in \mathbf{R} \mid x \neq-3, x \neq-4\} \) (d) \( \{x \in \mathbf{R} \mid x \neq-2, x \neq-6\} \)

To solve for the domain of the function \( y=\frac{x-3}{x^{2}-7 x+12} \), we first need to identify where the function is undefined. The denominator \( x^2 - 7x + 12 \) can be factored into \( (x-3)(x-4) \). This means the function is undefined at \( x=3 \) and \( x=4 \). Therefore, the domain includes all real numbers except for these two values, which is represented by: \[ \{x \in \mathbf{R} \mid x \neq 3, x \neq 4\} \] Now, isn’t math fascinating? The way it helps us identify where a function can “live” or “thrive” without hiccups is quite powerful! If you think about it, we have to keep an eye on the denominator to avoid getting involved with division by zero. When we recognize these key points, it allows us to work with functions smoothly. And speaking of avoiding pitfalls, remember that when you're dealing with rational functions, the zeros of the denominator are crucial to identifying restrictions on the domain. For instance, mixing up the values that make the denominator zero is a common mistake. Always start by factoring and solving the denominator; this ensures you don’t miss potential exclusions from the domain. It’s a quick check that saves a lot of future head-scratching moments!