https://file.aiquickdraw.com/mathbot/file/17264717640105fohqks5.jpg

The function provided in the image is:

\[
f(x, y) = \sqrt{x^2 - 2y}
\]

### 1. **Domain**:
The domain of a function refers to all the possible input values (in this case, \(x\) and \(y\)) for which the function is defined.

For a square root function, the expression inside the square root must be non-negative. Therefore, we need:

\[
x^2 - 2y \geq 0
\]

Solving this inequality:

\[
x^2 \geq 2y
\]

\[
y \leq \frac{x^2}{2}
\]

Thus, the domain of the function is all pairs \((x, y)\) where \( y \leq \frac{x^2}{2} \).

### 2. **Range**:
The range of a function is the set of all possible output values.

Since the function is a square root, the output of the square root is always non-negative. Therefore, the range of the function is:

\[
f(x, y) \geq 0
\]

Thus, the range of the function is all non-negative real numbers \([0, \infty)\).

### 3. **Sketching the Domain**:
To sketch the domain, you would plot the region \( y \leq \frac{x^2}{2} \) in the \(xy\)-plane. This represents the area below or on the parabola \( y = \frac{x^2}{2} \).

- The curve \( y = \frac{x^2}{2} \) is a parabola that opens upwards.
- The domain includes all points \((x, y)\) below or on this curve.

Do you need further explanation or help sketching the domain?

### Related Questions:
1. How do you determine the domain of other types of functions (like logarithmic or rational functions)?
2. What happens if the expression inside the square root was \(x^2 + 2y\) instead of \(x^2 - 2y\)?
3. Can a function have an empty domain? What would that mean?
4. How do you determine the range for functions other than square roots?
5. How do transformations (like scaling or translating) affect the domain and range of functions?

### Tip:
Always check for restrictions in a function (such as square roots or divisions) when determining the domain!

Find the domain and range and sketch the domain of the following function: f(x, y) = sqrt(x^2 - 2y).

Find the domain and range of the function f(x, y) = sqrt(x^2 - 2y) and learn how to sketch its domain. This problem explores square root functions and inequalities and is suitable for students in Grades 11-12.

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