https://file.aiquickdraw.com/mathbot/file/1726622008391iwe9gkxi.jpg

The function presented in the question is:

\[
f(x) = \frac{x^2 - 6x + 3}{x - 6}
\]

To determine the domain on which this function is continuous, we need to consider where it is *not* continuous, which typically occurs when the denominator equals zero, making the function undefined.

In this case, the denominator is \(x - 6\), so the function will be undefined when:

\[
x - 6 = 0 \implies x = 6
\]

Thus, the function is not continuous at \(x = 6\). Therefore, the function is continuous for all real numbers except \(x = 6\).

**Domain:** \( (-\infty, 6) \cup (6, \infty) \)

Would you like further details or have any questions? Here are some related questions to expand on this topic:

1. How do you find discontinuities for rational functions in general?
2. Can a function be continuous at a point where the denominator is zero?
3. How does the factorization of the numerator affect the continuity of the function?
4. What is the difference between a removable and non-removable discontinuity?
5. How can limits help us determine the behavior of a function near discontinuities?

**Tip:** Always check both the numerator and denominator when determining where a rational function is continuous.

On what domain is the function f(x) = (x^2 - 6x + 3) / (x - 6) continuous?

Explore how to find the domain of continuity for the rational function f(x) = (x^2 - 6x + 3) / (x - 6). Learn about the importance of checking when the denominator equals zero and understand continuity and discontinuities in rational functions. Suitable for high school calculus students.

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