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The domain and range are fundamental concepts when dealing with functions. The **domain** of a function is the set of all possible inputs (or x-values) that will not lead to undefined situations like division by zero or a negative square root. For the function given, \( f(x, y)=\sqrt{4-x^{2}-4 y^{2}} \), we need to ensure that the expression under the square root is non-negative.We start with the inequality:\[4 - x^2 - 4y^2 \geq 0\]Solving this inequality, we arrive at:\[x^2 + 4y^2 \leq 4\]This inequality describes a region in the XY-plane, specifically an ellipse centered at the origin with a horizontal semi-major axis of 2 and a vertical semi-minor axis of 1. The domain, therefore, consists of all (x, y) pairs inside and on this ellipse.The **range** of the function includes all the potential outputs, which are the resulting values of the function. For the square root, the smallest value is 0 (when \(x^2 + 4y^2 = 4\)) and the largest is 2 (when \(x = 0\) and \(y = 0\)). Thus, the range is:\[[0, 2]\]

Multivariable functions are functions with more than one input variable. They are an extension of single-variable functions to higher dimensions. In our function \( f(x, y)=\sqrt{4-x^{2}-4 y^{2}} \), the variables are x and y.These functions can be visualized as surfaces in a 3-dimensional space. Each input pair (x, y) corresponds to a point on this surface. For the given function, the surface is part of a 3D shape restricted by the inequality \(x^2 + 4y^2 \leq 4\).Understanding multivariable functions involves:

• **Domain:** Knowing where the function is defined.
• **Range:** Recognizing the possible output values.
• **Graphically:** Visualizing the function can aid comprehension. Here, imagine parts of a dome centered at the origin, reflecting the values affected by both x and y.

Inequalities often define the domain of a function. In this case, the inequality \( 4 - x^2 - 4y^2 \geq 0 \) ensures the expression inside the square root is never negative.To solve it, we rearranged it successfully to \( x^2 + 4y^2 \leq 4 \). Solutions to this inequality include all x and y pairs that are within or on an ellipse with axes scaling according to sqrt and absolute limits of the function.Inequalities guide us:

• **Non-negativity:** Ensuring the expression inside functions like square roots remains positive or zero.
• **Visual Representation:** Helps in understanding where the function is appropriately defined.

Coordinate geometry, also known as analytic geometry, allows us to represent and solve geometric problems using coordinates and algebra. For the function \( f(x, y)=\sqrt{4-x^{2}-4y^{2}} \), coordinate geometry lets us examine the domain's shape and size on the xy-plane.The inequality \( x^2 + 4y^2 \leq 4 \) describes an elliptical region. By interpreting coordinates and dimensions, it's understood as:

• **Ellipse:** Stretched along the x-axis and contracted along the y-axis.
• **Center:** Located at the origin (0,0) of the coordinate plane.
• **Axes:** Semi-major axis is 2 along x, and semi-minor axis is 1 along y, distinguishing it from a circle.

Coordinate geometry enables us to express relationships succinctly, blending algebra and geometry into a unified study to explore shapes and figures.