To find \(\left(\frac{f}{g}\right)(x)\), we divide \(f(x)\) by \(g(x)\). The given functions are:

- \(f(x) = x^2 - 5x + 6\)
- \(g(x) = x - 3\)

Now, we compute:

\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x^2 - 5x + 6}{x - 3}
\]

We can simplify this by factoring the numerator \(x^2 - 5x + 6\):

\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\]

Thus, we get:

\[
\left(\frac{f}{g}\right)(x) = \frac{(x - 2)(x - 3)}{x - 3}
\]

For \(x \neq 3\), the \((x - 3)\) terms cancel out, and we are left with:

\[
\left(\frac{f}{g}\right)(x) = x - 2 \quad \text{for} \, x \neq 3
\]

So, the simplified function is:

\[
\left(\frac{f}{g}\right)(x) = x - 2 \quad \text{with} \, x \neq 3
\]

Would you like further details or explanations?

Here are 5 related questions to expand on this:
1. What happens to the function when \(x = 3\)?
2. How do you factor other quadratic expressions like \(x^2 - 5x + 6\)?
3. What is the domain of the function \(\left(\frac{f}{g}\right)(x)\)?
4. Can you graph the function \(\left(\frac{f}{g}\right)(x)\) and explain its key features?
5. How do you apply division of functions to real-life scenarios?

**Tip:** Always check for undefined points (like \(x = 3\) here) when dividing functions!

Learn how to simplify the rational function (f/g)(x) where f(x) = x^2 - 5x + 6 and g(x) = x - 3. This solution covers algebraic techniques like factoring and discusses the domain of the function. Suitable for students in Grades 9-10.

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