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).

Simplify the rational function f(x) = (-3x^2 + 6x)/(2x^2) step by step. This problem focuses on factoring, canceling common terms, and rewriting expressions. Learn about rational functions, algebraic simplifications, and function domain restrictions. Suitable for students in Grades 9-12.

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