Final answer:
The domain of the function f(x,y,z) = ln(16 - 4x² - 4y² - z²) is the set of points inside an ellipsoid centered at the origin. This is visualized by sketching a cross-sectional ellipse in the xy-plane and extending the domain in the z-direction up to ±4.
Step-by-step explanation:
To find the domain of the function f(x,y,z) = ln(16 - 4x² - 4y² - z²), we must consider the argument of the natural logarithm function. The natural logarithm function, ln(u), is defined only when u > 0. Therefore, for f(x,y,z) to be defined, we need 16 - 4x² - 4y² - z² > 0. Rearranging terms, we get:
4x² + 4y² + z² < 16
Dividing all terms by 4, we simplify to:
x² + y² + (z²/4) < 4
This inequality describes the interior of an ellipsoid centered at the origin with radii 2 along the x-axis and y-axis, and 4 along the z-axis. The domain of f is therefore the set of all points (x, y, z) inside this ellipsoid.
To sketch the domain, we would draw an ellipse in the xy-plane representing the cross-section of the ellipsoid and indicate with shading or another method that the domain extends in the positive and negative z-direction up to the value of ±4, forming a three-dimensional ellipsoid.