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.