To determine the domain $M_f$ of the function $f(x) = \ln(x + 4) + \frac{1}{\sqrt{6 - x}}$, we need to identify the values of $x$ for which the function is defined.

1. Analyze the logarithmic term $\ln(x + 4)$:
   The argument of the natural logarithm, $x + 4$, must be positive for $f(x)$ to be defined. Thus: $x + 4 > 0 \Rightarrow x > -4.$

2. Analyze the square root in the denominator $\frac{1}{\sqrt{6 - x}}$:
   For the square root to be defined and the fraction to be finite, $6 - x$ must be positive. Therefore: $6 - x > 0 \Rightarrow x < 6.$

3. Combine the conditions:
   The domain $M_f$ of $f(x)$ is given by the intersection of these two inequalities: $-4 < x < 6.$

Therefore, the domain of $f(x)$ is: $M_f = \{ x \in \mathbb{R} \mid -4 < x < 6 \}.$

Would you like a more detailed breakdown of each step?

Here are five related questions for further exploration:

1. How would the domain change if we had $\ln(x + 2)$ instead of $\ln(x + 4)$?
2. What if the term in the denominator was $\frac{1}{6 - x}$ without the square root?
3. How would the domain be affected if there was an additional absolute value function around $x + 4$?
4. What if $f(x)$ included an additional term like $\sqrt{x - 5}$?
5. Can you graph the domain $M_f$ on a number line for visualization?

Tip: When finding the domain, check restrictions for each individual part of the function before combining them.