Math Problem Statement

what kind of function is \displaylines23x4x2{\displaylines2-3x\\ 4x^2}

Solution

The expression you've provided, interpreted as:

f(x)=23x4x2f(x) = \frac{2 - 3x}{4x^2}

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

Characteristics of this function:

  1. Numerator (P(x)): 23x2 - 3x, a linear polynomial.
  2. Denominator (Q(x)): 4x24x^2, a quadratic polynomial.

Domain:

The function is undefined where the denominator is zero. Thus, for this function, we must have 4x204x^2 \neq 0, or equivalently x0x \neq 0. The domain is all real numbers except x=0x = 0.

Asymptotic Behavior:

  • Vertical Asymptote: The function has a vertical asymptote at x=0x = 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=0y = 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=0x = 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.

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Polynomials
Asymptotes
Domain of Functions

Formulas

Rational function form: f(x) = \frac{P(x)}{Q(x)}
Asymptotes: Horizontal asymptote y = 0 when degree of Q(x) > P(x)

Theorems

Vertical Asymptote Theorem: A vertical asymptote occurs where the denominator equals zero.
Horizontal Asymptote Rule: If the degree of the denominator is higher than the numerator, the horizontal asymptote is y = 0.

Suitable Grade Level

Grades 10-12