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To find the **domain** of the function \( f(x) = \frac{5x + 1}{x^2 - 6x + 8} \), we need to determine when the denominator is not equal to zero because division by zero is undefined.

The denominator is:

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

We solve the equation \( x^2 - 6x + 8 = 0 \) by factoring:

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

Thus, the denominator equals zero when \( x = 2 \) or \( x = 4 \).

Therefore, the domain of the function is all real numbers except \( x = 2 \) and \( x = 4 \), or in interval notation:

\[
(-\infty, 2) \cup (2, 4) \cup (4, \infty)
\]

Would you like more details on the process? Let me know if you have any other questions!

### Related Questions:
1. How do we find the vertical asymptotes for this function?
2. What are the horizontal asymptotes for this rational function?
3. How does the graph of this function look?
4. What is the behavior of the function near \(x = 2\) and \(x = 4\)?
5. How can we analyze the continuity of this function?

### Tip:
When finding the domain of a rational function, always check when the denominator becomes zero as these values are excluded from the domain.

Learn how to find the domain of the rational function f(x) = (5x + 1) / (x^2 - 6x + 8). This problem covers algebraic concepts like factoring quadratic equations to identify undefined points, suitable for Grades 9-11 students.

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