https://file.aiquickdraw.com/mathbot/file/1728106691027x1v3b1tk.jpg

Let's solve for the domain of each of the functions given in the image (problems 5-8).

### Problem 8:
The function is:
\[
f(x, y) = \sqrt{y} + \sqrt{25 - x^2 - y^2}
\]

#### Step 1: Analyze the square root functions

1. **First square root: \( \sqrt{y} \)**  
   For this term to be defined, we need:
   \[
   y \geq 0
   \]
   
2. **Second square root: \( \sqrt{25 - x^2 - y^2} \)**  
   For this term to be defined, we require:
   \[
   25 - x^2 - y^2 \geq 0
   \]
   This inequality can be rearranged as:
   \[
   x^2 + y^2 \leq 25
   \]
   This describes the interior of a circle with radius 5 centered at the origin \((0, 0)\) on the \(xy\)-plane.

#### Step 2: Combine the conditions

To find the domain, we need both conditions to hold:
- \( y \geq 0 \) (the first square root requires this).
- \( x^2 + y^2 \leq 25 \) (the second square root defines the boundary of a circle).

Thus, the domain is the upper half (including the boundary) of the circle with radius 5 centered at the origin. This means that the function is defined for any point \((x, y)\) where:
- \( y \geq 0 \),
- \( x^2 + y^2 \leq 25 \).

This corresponds to the set of points inside the circle (including the boundary) where \( y \) is non-negative.

#### Step 3: Sketch the domain

- The domain is the upper half of a circle with radius 5 centered at the origin \((0, 0)\).
- The boundary of the circle is \( x^2 + y^2 = 25 \), and the domain includes the boundary.
- Only points with \( y \geq 0 \) are included.

If you'd like, I can provide a detailed sketch or graph of the domain.

---

Would you like any further details or a graph of this domain? 

Here are some related questions:
1. What is the domain of \( f(x, y) = \sqrt{1 - x^2} + \sqrt{1 - y^2} \)?
2. How do you find the domain of a logarithmic function like \( \ln(9 - x^2 - y^2) \)?
3. What is the domain of \( f(x, y) = \sqrt{16 - x^2 - y^2} \)?
4. Can you explain how to find the range of a function like this?
5. What changes in the domain if the second term is negative \( -\sqrt{25 - x^2 - y^2} \)?

**Tip:** Always start by analyzing each component (like square roots or logs) individually to identify restrictions for each term, then combine them to get the overall domain.

Find and sketch the domain of the function f(x, y) = sqrt(y) + sqrt(25 - x^2 - y^2).

Learn how to find and sketch the domain of the function f(x, y) = sqrt(y) + sqrt(25 - x^2 - y^2). This problem covers key mathematical concepts, including square roots, circle equations, and domain restrictions. Suitable for students in Grades 10-12.

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