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.