We are asked to describe the graph of the function $f(x, y) = \sqrt{16 - x^2 - y^2}$.

1. Problem Description

We are asked to describe the graph of the function $f(x, y) = \sqrt{16 - x^2 - y^2}$.

2. Solution Steps

The function is $f(x, y) = \sqrt{16 - x^2 - y^2}$. Let $z = f(x, y)$. Then we have

$z = \sqrt{16 - x^2 - y^2}$.

Squaring both sides, we get

$z^2 = 16 - x^2 - y^2$.

Rearranging the terms, we obtain

$x^2 + y^2 + z^2 = 16$.

This is the equation of a sphere centered at the origin $(0, 0, 0)$ with radius $r = \sqrt{16} = 4$.

However, since $z = f(x, y) = \sqrt{16 - x^2 - y^2}$, we must have $z \geq 0$.

Therefore, the graph of $f(x, y) = \sqrt{16 - x^2 - y^2}$ is the upper half of the sphere $x^2 + y^2 + z^2 = 16$.

3. Final Answer

The graph is the upper hemisphere of radius 4 centered at the origin. Specifically, it's the set of points $(x, y, z)$ such that $x^2 + y^2 + z^2 = 16$ and $z \geq 0$.