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The domain of a function refers to all the possible inputs, which are typically the x-values. For a function to make sense mathematically, it must only include inputs that keep the expression valid. In the case of the square root function, like the one in our example: \(h(x) = \sqrt{4-x^2}\), the expression under the square root must be non-negative. We cannot take the square root of a negative value and remain in the realm of real numbers.To find the domain here, set up the inequality:

• \(4 - x^2 \geq 0\)
• This simplifies to \(x^2 \leq 4\)
• Solving gives \(-2 \leq x \leq 2\)

Hence, the domain of the function includes all x-values from -2 to 2. Writing it in interval notation, it is: \([-2, 2]\). This range ensures that the function can operate without causing any mathematical errors due to negative square root inputs.

The range of a function encompasses all possible output values, often represented by y-values. For the function \(h(x) = \sqrt{4-x^2}\), since it's a square root function, outputs are restricted to non-negative numbers.The critical point of the range is determined by the maximum value inside the square root. Plug in different domain values to see how the output behaves:

• At \(x = 0\), \(h(0) = \sqrt{4} = 2\). Therefore, 2 is the maximum value the function reaches.
• At the ends \(x = -2\) and \(x = 2\), \(h(-2) = \sqrt{0} = 0\) and similarly for \(h(2) = 0\).

Thus, the range of this function is \([0, 2]\). The outputs smoothly transition from 0, at the domain ends, up to 2 at the peak.

The square root function can initially seem tricky, but it becomes predictable once you get to know it. It always produces non-negative results because you can't have the square root of a negative number in the set of real numbers. This function reflects a semi-circular nature when paired with a squared expression under the root, such as \(\sqrt{4-x^2}\).A few key characteristics of the square root:

• It is only defined for non-negative values, ensuring real number outputs.
• The output grows smaller as you move further from the center (or peak) of the range.
• It's often used in various mathematical applications such as calculating distances, physics problems, and probability.

Understanding this function helps to predict its behaviour: the expression \(4-x^2\) essentially creates a reflection of a half-circle when graphed.

Graph sketching is a visual representation of the function's behaviour over its domain. With the function \(h(x) = \sqrt{4-x^2}\), graphing can offer an intuitive insight into how the outputs change as inputs vary. This particular function forms a semi-circle shape, so let's see how the graph should look.Steps to sketch the graph:

• Identify the critical points: with the given function, essential points are \((-2, 0)\), \(0, 2\), and \(2, 0)\).
• Start sketching from these endpoints. The line between each point forms the curved boundary of the semi-circle.
• The highest point on the graph will be \( (0, 2) \) which forms the peak of your semi-circle.

Overall, you should have a semi-circle above the x-axis with endpoints at the domain's limits at \(-2\) and \(2\), with a maximal height of \(2\) in the middle. This represents the solution comprehensively, showing all critical points.