https://file.aiquickdraw.com/mathbot/file/17265836996480lenlkwy.jpg

Let's go through each function in the image and find its domain:

1. **\( y = 4x + 3 \)** (Part a):
   - This is a linear function, and linear functions are defined for all real numbers.
   - **Domain**: \( (-\infty, \infty) \)

2. **\( y = \frac{7}{x} \)** (Part b):
   - This is a rational function. The only restriction is that \( x \neq 0 \) because division by zero is undefined.
   - **Domain**: \( (-\infty, 0) \cup (0, \infty) \)

3. **\( y = \frac{1}{x - 5} \)** (Part c):
   - Another rational function. The restriction is that \( x \neq 5 \), since division by zero is undefined.
   - **Domain**: \( (-\infty, 5) \cup (5, \infty) \)

4. **\( y = \frac{3}{x + 8} \)** (Part d):
   - This is a rational function with the restriction \( x \neq -8 \) to avoid division by zero.
   - **Domain**: \( (-\infty, -8) \cup (-8, \infty) \)

5. **\( y = \sqrt{x - 2} \)** (Part e):
   - This is a square root function, and the expression inside the square root must be non-negative. So, \( x - 2 \geq 0 \), meaning \( x \geq 2 \).
   - **Domain**: \( [2, \infty) \)

6. **\( y = 2x^2 + 3 \)** (Part f):
   - This is a quadratic function, which is defined for all real numbers.
   - **Domain**: \( (-\infty, \infty) \)

7. **\( y = \frac{2}{x + 5} \)** (Part g):
   - This is a rational function with the restriction \( x \neq -5 \) to avoid division by zero.
   - **Domain**: \( (-\infty, -5) \cup (-5, \infty) \)

8. **\( y = \sqrt{x + 6} \)** (Part h):
   - This is a square root function. The expression inside the square root must be non-negative, so \( x + 6 \geq 0 \), meaning \( x \geq -6 \).
   - **Domain**: \( [-6, \infty) \)

### Summary of Domains:
- (a) \( (-\infty, \infty) \)
- (b) \( (-\infty, 0) \cup (0, \infty) \)
- (c) \( (-\infty, 5) \cup (5, \infty) \)
- (d) \( (-\infty, -8) \cup (-8, \infty) \)
- (e) \( [2, \infty) \)
- (f) \( (-\infty, \infty) \)
- (g) \( (-\infty, -5) \cup (-5, \infty) \)
- (h) \( [-6, \infty) \)

Would you like more details on any of these functions?

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Here are 5 follow-up questions:
1. How do you find the domain of a rational function?
2. What happens to the domain when a square root function is involved?
3. Can the domain of a linear function ever have restrictions?
4. How would you graph the domain of \( y = \frac{1}{x-5} \)?
5. What are some real-world applications of domain restrictions in functions?

**Tip**: For rational functions, always look out for values of \(x\) that make the denominator zero to avoid undefined values.

Learn how to find the domain of different types of functions, including linear, rational, and square root functions. Understand key algebraic concepts such as avoiding division by zero and ensuring non-negative values for square roots. Suitable for high school students (Grades 9-12).

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