Math Problem Statement
what kind of function is
Solution
The expression you've provided, interpreted as:
is a rational function. A rational function is a function of the form , where both and are polynomials and .
Characteristics of this function:
- Numerator (P(x)): , a linear polynomial.
- Denominator (Q(x)): , a quadratic polynomial.
Domain:
The function is undefined where the denominator is zero. Thus, for this function, we must have , or equivalently . The domain is all real numbers except .
Asymptotic Behavior:
- Vertical Asymptote: The function has a vertical asymptote at 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 .
Would you like more details or have any other questions?
Related Questions:
- What is the domain of a general rational function?
- How do you determine the vertical asymptotes of a rational function?
- How do you find the horizontal or oblique asymptotes for rational functions?
- What is the behavior of the function near ?
- 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
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