To find the domain of the function:

$f(x) = \sqrt{x + 7} - \frac{\sqrt{5 - x}}{x}$

we need to ensure that the function is defined at all points. There are two primary conditions we must check:

1. Square roots:

• The square root function $\sqrt{u}$ is only defined for $u \geq 0$. Thus, we have two conditions for the square roots in the expression:
  1. $x + 7 \geq 0 \Rightarrow x \geq -7$
  2. $5 - x \geq 0 \Rightarrow x \leq 5$

2. Denominator:

• The function has a denominator $x$ in the second term, and division by zero is undefined. Therefore, we must also have: $x \neq 0$

Summary of conditions:

• $x \geq -7$ (from $\sqrt{x+7}$)
• $x \leq 5$ (from $\sqrt{5 - x}$)
• $x \neq 0$ (to avoid division by zero)

Final domain:

The domain is the interval where all three conditions are satisfied:

• $x \in [-7, 5]$, excluding 0.

Thus, the domain of the function is:

$\boxed{[-7, 0) \cup (0, 5]}$

Would you like more details or explanations on any step?

Here are 5 related questions to explore further:

1. How do we handle the domain when both square roots and rational functions are present?
2. What happens to the function near the discontinuity at $x = 0$?
3. How would the domain change if the square root functions were replaced with different powers?
4. How does the range of this function compare to its domain?
5. Can this function have any vertical or horizontal asymptotes?

Tip: When working with square roots, always check if the expression inside can become negative—it affects the domain!