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The document contains several math problems. I'll walk you through how to solve each one.

### 1. Domain of Functions

For each function, we need to determine the domain, which is the set of all possible \( x \)-values for which the function is defined.

#### (a) \( f(x) = \frac{3}{x - 2} \)

- The function is undefined when the denominator is zero.
- Set the denominator equal to zero and solve: \( x - 2 = 0 \), so \( x = 2 \).
- The domain is all real numbers except \( x = 2 \).

**Domain:** \( (-\infty, 2) \cup (2, \infty) \)

#### (b) \( g(x) = 5x^2 - x - 1 \)

- This is a polynomial function, which is defined for all real numbers.
  
**Domain:** \( (-\infty, \infty) \)

#### (c) \( h(x) = \frac{6x - 2}{x^3 - 3x} \)

- The function is undefined when the denominator is zero.
- Factor the denominator: \( x^3 - 3x = x(x^2 - 3) \).
- Solve \( x(x^2 - 3) = 0 \). The solutions are \( x = 0 \) and \( x = \pm\sqrt{3} \).
- The domain is all real numbers except \( x = 0, \pm\sqrt{3} \).

**Domain:** \( (-\infty, -\sqrt{3}) \cup (-\sqrt{3}, 0) \cup (0, \sqrt{3}) \cup (\sqrt{3}, \infty) \)

### 2. Graphing a Piecewise-Defined Function

You need to graph the following piecewise-defined function:

\[
f(x) = 
\begin{cases} 
x - 1, & \text{if } -4 \leq x < 1 \\
3, & \text{if } 1 \leq x \leq 4 
\end{cases}
\]

- For \( -4 \leq x < 1 \), graph the line \( y = x - 1 \) but only for \( x \) in this range.
- For \( 1 \leq x \leq 4 \), draw a horizontal line \( y = 3 \).

### 3. Difference Quotient for \( f(x) = 3x^2 - 2x + 1 \)

The difference quotient is:

\[
\frac{f(a+h) - f(a)}{h}
\]

- Compute \( f(a+h) = 3(a+h)^2 - 2(a+h) + 1 \).
- Expand and simplify.
- Subtract \( f(a) = 3a^2 - 2a + 1 \) from \( f(a+h) \).
- Divide the result by \( h \).

### 4. Difference Quotient for \( g(x) = \frac{3}{x-1} \)

- Compute \( g(a+h) = \frac{3}{a+h-1} \).
- The difference quotient is then \( \frac{\frac{3}{a+h-1} - \frac{3}{a-1}}{h} \).
- Simplify by finding a common denominator.

Would you like detailed steps for any specific problem?

### Related Questions:
1. How do you simplify the difference quotient expression after substitution?
2. What is the significance of undefined points in a function's domain?
3. How do you determine the correct range for a piecewise function?
4. What happens to the graph of a piecewise function at the boundaries of its intervals?
5. How do you graph a quadratic function?

**Tip:** Always check for undefined points when determining the domain of a function!

Learn how to solve equations involving domain of functions, graphing piecewise functions, and computing difference quotient. This detailed guide covers functions such as f(x) = 3/(x - 1), g(x) = 5x^2 - x - 1, and h(x) = (6x - 2)/(x^3 - 3x). Suitable for high school students seeking in-depth explanations and examples.

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