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.