f(x,y)=sqrt(9-x^2-y). Explain how to calculate the domain and range


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.

f(x, y) = sqrt(9 - x^2 - y). Explain how to calculate the domain and range.

Learn how to find the domain and range of the multivariable function f(x, y) = sqrt(9 - x^2 - y). This problem covers key concepts in domain and range for square root functions and is ideal for high school students in Grades 10-12.

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