To calculate the domain and range of the function $f(x, y) = \sqrt{9 - x^2 - y}$, let's break it down step by step.

1. Domain:

The domain of a function refers to all possible input values (in this case, pairs $(x, y)$) that make the function valid. For a square root function to be defined, the expression inside the square root must be non-negative because the square root of a negative number is not a real number.

So, for $f(x, y) = \sqrt{9 - x^2 - y}$ to be valid, the expression inside the square root must satisfy:

$9 - x^2 - y \geq 0$

Rearranging this inequality:

$y \leq 9 - x^2$

This inequality tells us the possible values for $y$ given any $x$. So, for each $x$, the corresponding $y$ must be less than or equal to $9 - x^2$. The value of $x$ must also satisfy $-3 \leq x \leq 3$ because $x^2 \leq 9$.

Thus, the domain is all pairs $(x, y)$ such that:

$-3 \leq x \leq 3 \quad \text{and} \quad y \leq 9 - x^2$

2. Range:

The range of a function refers to all possible output values (the values of $f(x, y)$).

Since the function is a square root, the output of $f(x, y)$ is always non-negative. Additionally, the value of the square root reaches its maximum when the expression inside the square root is maximized. The maximum value of $9 - x^2 - y$ occurs when $x = 0$ and $y = 0$, which gives:

$f(0, 0) = \sqrt{9 - 0^2 - 0} = \sqrt{9} = 3$

Thus, the range of the function is:

$[0, 3]$

Summary:

• The domain is all $(x, y)$ such that $-3 \leq x \leq 3$ and $y \leq 9 - x^2$.
• The range is all real values $f(x, y) \in [0, 3]$.

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Do you want further details on any part of this explanation? Here are five questions that build on this concept:

1. How does the shape of the domain look in the xy-plane?
2. What happens if the expression inside the square root is negative?
3. Can the range of a square root function ever include negative values?
4. How would the domain change if the function were $f(x, y) = \sqrt{9 - x^2 - y^2}$?
5. Can we extend this reasoning to functions involving cube roots?

Tip: Always check the inside of the square root for non-negative values when calculating the domain of square root functions.