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?

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.