The function you provided is:

\[
F(x) = \sqrt{4 - x^2}
\]

### Domain of the function:
To determine the domain of the function \( F(x) \), we need to ensure that the expression inside the square root is non-negative, since the square root of a negative number is not defined in the set of real numbers.

\[
4 - x^2 \geq 0
\]

Solving this inequality:

\[
4 \geq x^2 \quad \Rightarrow \quad -2 \leq x \leq 2
\]

Thus, the domain of \( F(x) \) is \( x \in [-2, 2] \).

### Range of the function:
Since the function involves a square root, the output will always be non-negative. The maximum value occurs when \( x = 0 \), which gives:

\[
F(0) = \sqrt{4 - 0^2} = \sqrt{4} = 2
\]

The minimum value occurs when \( x = \pm 2 \), which gives:

\[
F(\pm 2) = \sqrt{4 - 2^2} = \sqrt{0} = 0
\]

Thus, the range of the function is \( [0, 2] \).

### Conclusion:
- **Domain**: \( [-2, 2] \)
- **Range**: \( [0, 2] \)

Would you like more details on this function, such as its graph or properties?

Here are 5 related questions you could explore:
1. How do you graph the function \( F(x) = \sqrt{4 - x^2} \)?
2. What is the derivative of \( F(x) = \sqrt{4 - x^2} \)?
3. How do you find the area under the curve of \( F(x) \) between \( -2 \) and \( 2 \)?
4. Can you describe the geometric shape represented by \( F(x) = \sqrt{4 - x^2} \)?
5. How would the function change if you modified the equation to \( \sqrt{9 - x^2} \)?

**Tip**: Always check the domain restrictions for square roots and rational functions to ensure the expression is valid for all inputs.

Explore the domain and range of the function F(x) = \sqrt{4 - x^2}. Learn how to solve for the domain by analyzing the inequality and determine the range based on the behavior of square root functions. This detailed guide is suitable for students in Grades 9-11.

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