1b **Problem Statement:**Find and sketch the domain of the function \( f(x, y, z) = \sqrt{z - x^2 - y^2} \).**Explanation:**The function \( f(x, y, z) = \sqrt{z - x^2 - y^2} \) is defined only for non-negative values under the square root. Therefore, the expression \( z - x^2 - y^2 \) must be greater than or equal to zero:\[ z - x^2 - y^2 \geq 0 \]This can be rewritten as:\[ z \geq x^2 + y^2 \]The inequality \( z \geq x^2 + y^2 \) represents the region above the paraboloid in three-dimensional space. The paraboloid is centered along the z-axis and opens upward.**Graphical Representation:**To visualize the domain:1. **Identify the shape of the domain**: The inequality \( z \geq x^2 + y^2 \) describes the region above the paraboloid in 3D space, where each "slice" perpendicular to the z-axis is a circle (since \( x^2 + y^2 = z \) forms a circle in the xy-plane for constant z).2. **3D Visualization**: In a 3-dimensional coordinate system: - The xy-plane is horizontal. - The z-axis is vertical. - The paraboloid is the surface where \( z = x^2 + y^2 \).3. **Domain Interpretation**: The domain of \( f \) includes all points (x, y, z) where z is greater than or equal to \( x^2 + y^2 \).In conclusion, the domain of the function consists of the set of points that lie on or above the surface of the paraboloid defined by \( z = x^2 + y^2 \).

Name: 1b **Problem Statement:**Find and sketch the domain of the function \(
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: 1b **Problem Statement:**Find and sketch the domain of the function \( f(x, y, z) = \sqrt{z - x^2 - y^2} \).**Explanation:**The function \( f(x, y, z) = \sqrt{z - x^2 - y^2} \) is defined only for non-negative values under the square root. Therefore,