91Ó°ÊÓ

Understanding the domain and range of a multivariable function is crucial. The domain consists of all possible input values that make the function work without errors. For our function, defined as \(f(x, y) = \sqrt{4 - x^2 - y^2}\), we need the expression inside the square root to be non-negative (i.e., \(4 - x^2 - y^2 \geq 0\)). This constraint ensures that the function is real-valued.

Once we rearrange this condition, we identify that \(x^2 + y^2 \leq 4\). This describes a circle on the xy-plane with a radius of 2. Hence, the domain is all points inside or on this circle. It includes the boundary because the inequality is \(\leq\), not just less than.

Meanwhile, the range, representing all possible output values, is found by evaluating the square root function's limits. Here, values range from 0, when \(x^2 + y^2 = 4\), to 2 when both x and y are zero. Thus, the range is \([0, 2]\).

The square root function, often represented as \(\sqrt{x}\), is always non-negative because it reflects the principal square root. In our particular function, \(f(x, y) = \sqrt{4 - x^2 - y^2}\), it dictates the output of the function.

This influences both the domain and range of multivariable functions:

• **Domain:** Only defined for non-negative numbers under the root, hence our need to solve \(4 - x^2 - y^2 \geq 0\).
• **Range:** Square root values start from 0 and can go up to the square root of any positive number; in our case, the maximum is 2 when the input equals zero, i.e., \(4 - x^2 - y^2 = 4\).

It's important to remember that the square root influences how far the function can stretch vertically.

Inequalities, such as \(4 - x^2 - y^2 \geq 0\), help define constraints in problems involving multiple variables. Solving them typically establishes boundaries or conditions that must be satisfied.

For this function, the inequality forms a circle's equation, \(x^2 + y^2 \leq 4\), stating that points must lie within or on this circle. It is vital to correctly interpret this because:

• **Boundary inclusion:** The "equal to" part (\(\leq\)) includes points on the circle.
• **Graphical understanding:** Visually identifies limits for problem-solving in multivariable contexts.

Interpreting these inequalities accurately is a significant step toward finding a correct domain.

The circle equation \(x^2 + y^2 = r^2\) represents a circle centered at the origin on a coordinate plane with radius \(r\). In our problem, the function encapsulates expressing an inequality in the form of a circle.

Here's what happens: the inequality \(x^2 + y^2 \leq 4\) describes a circle. The function stays defined as long as the points \((x, y)\) remain inside or on this circle.

• **Center and radius:** Center of the circle is at the origin \((0, 0)\), and the radius is 2 (since \(4 = 2^2\)).
• **Boundary:** The boundary of the circle forms part of the domain because of the \(\leq\) condition.

This geometric perspective helps in visualizing multivariable domains, transforming tough algebra into insightful geometry.