1. **State the problem:** We are given the function $$f(x) = \frac{2x^2 - 1}{x^2 - 5x + 6}$$ and we want to analyze it. 2. **Factor the denominator:** The denominator is a quadratic expression. We factor it as: $$x^2 - 5x + 6 = (x - 2)(x - 3)$$ This tells us the function is undefined at $$x = 2$$ and $$x = 3$$ because division by zero is undefined. 3. **Analyze the numerator:** The numerator is $$2x^2 - 1$$ which cannot be factored nicely with integers, so we leave it as is. 4. **Domain:** The domain of $$f(x)$$ is all real numbers except $$x = 2$$ and $$x = 3$$. 5. **Simplify if possible:** Since numerator and denominator share no common factors, the function cannot be simplified further. 6. **Summary:** The function is $$f(x) = \frac{2x^2 - 1}{(x - 2)(x - 3)}$$ with domain $$x \neq 2, 3$$. This is the complete analysis of the given function.