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To find the domain of the given rational function:

\[
y = \frac{8 - 2x}{x^2 - 6x + 8}
\]

we need to determine when the denominator is equal to zero because a rational function is undefined when its denominator is zero.

### Step 1: Set the denominator equal to zero.

\[
x^2 - 6x + 8 = 0
\]

### Step 2: Solve the quadratic equation.

We can factor the quadratic equation:

\[
x^2 - 6x + 8 = (x - 4)(x - 2)
\]

### Step 3: Set each factor equal to zero.

\[
x - 4 = 0 \quad \text{or} \quad x - 2 = 0
\]

So, \( x = 4 \) and \( x = 2 \).

### Step 4: Exclude these values from the domain.

The rational function is undefined at \( x = 4 \) and \( x = 2 \) because they make the denominator zero. Therefore, the domain of the function is all real numbers except \( x = 4 \) and \( x = 2 \).

### Final Answer:
The domain is:

\[
\boxed{(-\infty, 2) \cup (2, 4) \cup (4, \infty)}
\]

Would you like any further explanation or details on any step? 

Here are 5 questions for expanding on this topic:
1. How do you factor other quadratic expressions?
2. Why is it important to find where the denominator is zero in rational functions?
3. What happens to the graph of a function at points where the denominator is zero?
4. How can you determine the behavior of the function near the points \( x = 4 \) and \( x = 2 \)?
5. Could there be any vertical asymptotes at \( x = 4 \) and \( x = 2 \)?

**Tip:** Always check both the numerator and denominator for common factors before factoring the quadratic, as this could simplify the function and the domain restrictions.

Find the domain of the rational function: y = (8 - 2x) / (x^2 - 6x + 8)

Find the domain of the rational function y = (8 - 2x) / (x^2 - 6x + 8). Learn how to determine where the denominator is zero and how to exclude those values from the domain. Suitable for students in Grades 9-12.

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