what kind of function is $$ {\displaylines2-3x\\ 4x^2} $$

The expression you've provided, interpreted as:

\[
f(x) = \frac{2 - 3x}{4x^2}
\]

is a **rational function**. A rational function is a function of the form $ \frac{P(x)}{Q(x)} $, where both $ P(x) $ and $ Q(x) $ are polynomials and $ Q(x) \neq 0 $.

### Characteristics of this function:
1. **Numerator (P(x))**: $ 2 - 3x $, a linear polynomial.
2. **Denominator (Q(x))**: $ 4x^2 $, a quadratic polynomial.

### Domain:
The function is undefined where the denominator is zero. Thus, for this function, we must have $ 4x^2 \neq 0 $, or equivalently $ x \neq 0 $. The domain is all real numbers except $ x = 0 $.

### Asymptotic Behavior:
- **Vertical Asymptote**: The function has a vertical asymptote at $ x = 0 $ because the denominator becomes zero there.
- **Horizontal Asymptote**: To find the horizontal asymptote, compare the degree of the numerator and denominator. Since the denominator has a higher degree (degree 2 vs. degree 1 in the numerator), the horizontal asymptote is $ y = 0 $.

Would you like more details or have any other questions?

### Related Questions:
1. What is the domain of a general rational function?
2. How do you determine the vertical asymptotes of a rational function?
3. How do you find the horizontal or oblique asymptotes for rational functions?
4. What is the behavior of the function near $ x = 0 $?
5. How do you graph rational functions?

### Tip:
When determining the domain of a rational function, always check where the denominator equals zero, as these points are excluded from the domain.

what kind of function is $$ {\displaylines 2-3x \\ 4x^2} $$

Explore the rational function f(x) = (2 - 3x) / (4x^2) with an in-depth analysis of its domain, vertical asymptote at x = 0, and horizontal asymptote at y = 0. This solution is designed for students in Grades 10-12, covering key algebraic concepts.

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