To complete the task for the function $F(x) = \sqrt{x + 3}$, we need to evaluate this function for integer values of $x$ that ensure $x + 3$ is a perfect square. This will avoid decimal results in $y = F(x)$.

Step-by-Step Process:

1. For $y = F(x) = \sqrt{x + 3}$ to be a whole number, $x + 3$ must be a perfect square. That is: $x + 3 = n^2$ where $n$ is a whole number.

2. Solving for $x$, we get: $x = n^2 - 3$ Now, we'll evaluate this expression for different values of $n$.

Example Calculations:

• When $n = 2$: $x = 2^2 - 3 = 4 - 3 = 1$ So, $F(1) = \sqrt{1 + 3} = \sqrt{4} = 2$.

• When $n = 3$: $x = 3^2 - 3 = 9 - 3 = 6$ So, $F(6) = \sqrt{6 + 3} = \sqrt{9} = 3$.

• When $n = 4$: $x = 4^2 - 3 = 16 - 3 = 13$ So, $F(13) = \sqrt{13 + 3} = \sqrt{16} = 4$.

General Expression:

Thus, for any $n$, the function $F(x)$ will return a whole number if $x = n^2 - 3$.

Do you have any questions about these steps or need further details?

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Here are 5 related questions to deepen your understanding:

1. What happens if $x + 3$ is not a perfect square? How does it affect $F(x)$?
2. How can we generalize this approach to functions involving square roots and different constants?
3. What is the domain of the function $F(x) = \sqrt{x + 3}$, and why does it matter?
4. Can we apply the same logic for cube roots or higher roots?
5. What would the graph of $F(x) = \sqrt{x + 3}$ look like?

Tip: Always check the domain of functions involving square roots to avoid non-real results.