We are asked to describe the graphs of the given functions $f(x, y)$. Specifically, we'll analyze problem 11: $f(x, y) = \sqrt{16 - x^2 - y^2}$.

1. Problem Description

We are asked to describe the graphs of the given functions $f(x, y)$.

Specifically, we'll analyze problem 11: $f(x, y) = \sqrt{16 - x^2 - y^2}$.

2. Solution Steps

First, let's write $z = f(x, y)$, so we have $z = \sqrt{16 - x^2 - y^2}$.

Square both sides to get $z^2 = 16 - x^2 - y^2$.

Rearrange the equation to $x^2 + y^2 + z^2 = 16$. This is the equation of a sphere with radius $\sqrt{16} = 4$, centered at the origin.

Since $z = \sqrt{16 - x^2 - y^2}$, $z$ must be non-negative, $z \geq 0$. This means we only have the top half of the sphere.

3. Final Answer

The graph of $f(x, y) = \sqrt{16 - x^2 - y^2}$ is the top half of the sphere with radius 4 centered at the origin.