The problem asks for the domain of the function:

$f(x) = \sqrt{x + 4} - \frac{\sqrt{8 - x}}{x}$

To find the domain of this function, we need to analyze two key parts:

1. Square roots: Both $\sqrt{x + 4}$ and $\sqrt{8 - x}$ require the arguments inside the square roots to be non-negative.

   • For $\sqrt{x + 4}$, we need $x + 4 \geq 0$, which simplifies to: $x \geq -4$
   • For $\sqrt{8 - x}$, we need $8 - x \geq 0$, which simplifies to: $x \leq 8$

2. Division by $x$: Since the function contains a division by $x$, $x \neq 0$ must also be satisfied to avoid division by zero.

Combining the conditions:

• From the square roots, we require: $-4 \leq x \leq 8$
• From the division, we exclude $x = 0$.

Domain:

Thus, the domain is the interval $[-4, 8]$, excluding 0. In interval notation, this is written as: $[-4, 0) \cup (0, 8]$

The answer you provided, $(-4, 0) \cup (0, 8]$, is incorrect because $-4$ should be included in the domain.

Would you like more details or have further questions?

Here are 5 related questions:

1. How do we find the domain of functions involving radicals?
2. What is the significance of excluding points where the denominator is zero?
3. How do we combine multiple conditions when finding a domain?
4. What is the difference between square roots and higher even roots in domain problems?
5. How would the domain change if we had a cube root instead?

Tip: Always check for both radical and division restrictions when determining the domain of a function.