To find the domain of the function $F(x) = \sqrt{x + 4}$, we need to ensure that the expression inside the square root is non-negative because the square root of a negative number is not defined in the set of real numbers.

So, we set up the inequality:

$x + 4 \geq 0$

Solving for $x$:

$x \geq -4$

Thus, the domain of the function is all real numbers $x$ such that $x \geq -4$. In interval notation, the domain is:

$\boxed{[-4, \infty)}$

Would you like more details or have any questions?

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Here are 5 related questions for further exploration:

1. What happens to the domain if the square root expression is in the denominator instead of the numerator?
2. How does the domain change if the expression inside the square root is $x - 4$?
3. What is the range of the function $F(x) = \sqrt{x + 4}$?
4. How would the domain be affected if the function were $F(x) = \sqrt{x^2 + 1}$?
5. How would you graph the function $F(x) = \sqrt{x + 4}$?

Tip: When finding the domain of square root functions, always check the expression inside the square root and ensure it is greater than or equal to zero!