The function is given by:

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

Let's simplify this expression:

1. Factor the numerator: $f(x) = \frac{3x(-x + 2)}{2x^2}$

2. Cancel out the common factor of $x$ from the numerator and denominator (assuming $x \neq 0$): $f(x) = \frac{3(-x + 2)}{2x}$

So, the simplified form of the function is: $f(x) = \frac{-3x + 6}{2x}$

We can split this expression into two terms: $f(x) = \frac{-3x}{2x} + \frac{6}{2x}$ Simplifying further: $f(x) = -\frac{3}{2} + \frac{3}{x}$

Thus, the simplified form of the function is: $f(x) = -\frac{3}{2} + \frac{3}{x}$

Would you like more details or have any questions?

Here are 5 related questions to expand on this:

1. What are the domain restrictions for the function $f(x)$?
2. How would you find the vertical asymptote of $f(x)$?
3. What is the behavior of $f(x)$ as $x \to \infty$ or $x \to -\infty$?
4. Can you compute the first derivative $f'(x)$ and interpret its meaning?
5. How does the graph of $f(x)$ look like? Where are the intercepts?

Tip: When simplifying rational functions, always check for any factors that can cancel out and determine if there are restrictions on the domain (e.g., values of $x$ that would make the denominator zero).