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The problem presents several functions and asks for two tasks:

1. **Find the domain of each function.**
2. **Calculate specific values for each function.**

Let's start with each function:

### Given Functions:
1. \( f(x) = \frac{3x - 5}{x^2 - 9} \)
2. \( g(x) = \sqrt{\frac{x}{x - 1}} \)
3. \( h(x) = |x - 2| - 4 \)
4. \( b(x) = x^4 + 4x + 1 \)
5. \( d(x) = x^2 + 2 \)
6. \( r(x) = \log_{10}(4x + 3) \)

### 1. Finding the Domain of Each Function:

**a) \( f(x) = \frac{3x - 5}{x^2 - 9} \)**

The domain is all real numbers except where the denominator is zero:
\[ x^2 - 9 = 0 \]
\[ (x - 3)(x + 3) = 0 \]
So, \( x = 3 \) and \( x = -3 \) are excluded from the domain.

**Domain:** \( \mathbb{R} \setminus \{-3, 3\} \)

**b) \( g(x) = \sqrt{\frac{x}{x - 1}} \)**

The expression inside the square root must be non-negative, and the denominator cannot be zero:
\[ \frac{x}{x - 1} \geq 0 \]

This inequality is satisfied when:
1. \( x > 1 \) 
2. \( x = 0 \) (which makes the entire fraction zero).

Thus, the domain excludes the interval \( 0 < x \leq 1 \).

**Domain:** \( (-\infty, 0] \cup (1, \infty) \)

**c) \( h(x) = |x - 2| - 4 \)**

Absolute value functions are defined for all real numbers.

**Domain:** \( \mathbb{R} \)

**d) \( b(x) = x^4 + 4x + 1 \)**

This is a polynomial, and polynomials are defined for all real numbers.

**Domain:** \( \mathbb{R} \)

**e) \( d(x) = x^2 + 2 \)**

This is also a polynomial, so it's defined for all real numbers.

**Domain:** \( \mathbb{R} \)

**f) \( r(x) = \log_{10}(4x + 3) \)**

The argument of the logarithm must be positive:
\[ 4x + 3 > 0 \]
\[ x > -\frac{3}{4} \]

**Domain:** \( \left( -\frac{3}{4}, \infty \right) \)

### 2. Calculating Specific Function Values:

Now, let's calculate the requested values:

- \( f(-1) \)
  \[
  f(-1) = \frac{3(-1) - 5}{(-1)^2 - 9} = \frac{-3 - 5}{1 - 9} = \frac{-8}{-8} = 1
  \]

- \( g(2) \)
  \[
  g(2) = \sqrt{\frac{2}{2 - 1}} = \sqrt{2}
  \]

- \( b(0) \)
  \[
  b(0) = 0^4 + 4(0) + 1 = 1
  \]

- \( r(-\frac{2}{3}) \)
  \[
  r\left(-\frac{2}{3}\right) = \log_{10}\left(4\left(-\frac{2}{3}\right) + 3\right) = \log_{10}\left(-\frac{8}{3} + 3\right) = \log_{10}\left(\frac{1}{3}\right) 
  \]

- \( h(5) \)
  \[
  h(5) = |5 - 2| - 4 = 3 - 4 = -1
  \]

- \( d(0) \)
  \[
  d(0) = 0^2 + 2 = 2
  \]

### Conclusion
The domain and values have been determined. If you have any specific questions or want detailed explanations on how to solve these problems, feel free to ask.

### Relative Questions:
1. How do you solve quadratic inequalities, like the one in \( g(x) \)?
2. What are the properties of the absolute value function, and how do they affect the domain?
3. Why is the domain of logarithmic functions restricted?
4. How does the behavior of polynomial functions differ from rational functions?
5. What is the process for determining when a function is undefined?

### Tip:
Always remember to check for points where the function is undefined, like zero denominators or negative arguments under even roots or logarithms.

Find the domain of the given functions and calculate specific values for each function.

This problem explores finding the domain for multiple types of functions including rational, square root, polynomial, and logarithmic functions. It also involves calculating specific values for these functions, covering key concepts such as inequalities, absolute value, and logarithms. Ideal for students in Grades 10-12.

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