Let's break down the domain finding process for each function.

1. Function $f(x) = \frac{x+6}{x^2 + 2x - 24}$:

   • The domain of a rational function excludes values of $x$ that make the denominator equal to zero.
   • Set the denominator equal to zero and solve for $x$: $x^2 + 2x - 24 = 0$
   • Factor the quadratic equation: $(x + 6)(x - 4) = 0$
   • So, $x = -6$ and $x = 4$ are the values that make the denominator zero.
   • Therefore, the domain of $f(x)$ excludes $x = -6$ and $x = 4$.
   • The domain is: $(-\infty, -6) \cup (-6, 4) \cup (4, \infty)$

2. Function $g(x) = \frac{x}{x^2 + 9}$:

   • Again, the domain excludes values of $x$ that make the denominator zero.
   • Set the denominator equal to zero: $x^2 + 9 = 0$
   • Solving for $x$, we get: $x^2 = -9$
   • This has no real solution because a square cannot be negative.
   • Therefore, the denominator never equals zero for real numbers, meaning the function is defined for all real numbers.
   • The domain is: $(-\infty, \infty)$

Would you like more details or further clarification?

Here are 5 related questions:

1. How do we handle undefined points in rational functions?
2. Can you explain how to factor quadratic expressions like $x^2 + 2x - 24$?
3. What happens