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Let's address the given problem step by step.

The function provided is:

\[
f(x) = \frac{x^2 - 2x + 6}{2x - 1}
\]

We are tasked with:
a. Finding the domain of \( f(x) \).  
b. Evaluating \( f(0) \) and \( f(-4) \).

### Part a: Finding the Domain of \( f(x) \)

The domain of a rational function is all real numbers except where the denominator equals zero, because division by zero is undefined.

We set the denominator equal to zero to find the values of \( x \) to exclude:

\[
2x - 1 = 0
\]
\[
x = \frac{1}{2}
\]

Thus, the function is undefined at \( x = \frac{1}{2} \).

**Domain:** All real numbers except \( x = \frac{1}{2} \), or in interval notation:
\[
\text{Domain: } (-\infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty)
\]

### Part b: Evaluating \( f(0) \) and \( f(-4) \)

1. **Evaluating \( f(0) \):**

Substitute \( x = 0 \) into the function:

\[
f(0) = \frac{0^2 - 2(0) + 6}{2(0) - 1} = \frac{6}{-1} = -6
\]

So, \( f(0) = -6 \).

2. **Evaluating \( f(-4) \):**

Substitute \( x = -4 \) into the function:

\[
f(-4) = \frac{(-4)^2 - 2(-4) + 6}{2(-4) - 1} = \frac{16 + 8 + 6}{-8 - 1} = \frac{30}{-9} = -\frac{10}{3}
\]

So, \( f(-4) = -\frac{10}{3} \).

### Summary:

- **Domain:** \( (-\infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty) \)
- \( f(0) = -6 \)
- \( f(-4) = -\frac{10}{3} \)

Let me know if you need further details or have any questions! Here are five related questions to expand on this topic:

1. What is the range of the function \( f(x) \)?
2. Can you simplify the function \( f(x) \) further by factoring the numerator?
3. How would you determine vertical and horizontal asymptotes for \( f(x) \)?
4. What happens to \( f(x) \) as \( x \) approaches infinity?
5. How would you graph \( f(x) \)?

**Tip:** Always check for undefined points in rational functions by looking at where the denominator equals zero!

Given the function f(x) = (x^2 - 2x + 6) / (2x - 1), find the domain of f. Evaluate f(0) and f(-4).

Explore how to find the domain of the rational function f(x) = (x^2 - 2x + 6) / (2x - 1) and evaluate it at specific points like f(0) and f(-4). Understand how to handle undefined values and perform function evaluations step-by-step. Suitable for students in Grades 9-12.

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