Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Functions
A rational function is a function that can be expressed as the ratio of two polynomials, typically in the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials. Understanding the behavior of rational functions is crucial for analyzing their graphs, particularly in identifying asymptotes.
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Asymptotes
Asymptotes are lines that a graph approaches but never touches. There are three types: vertical asymptotes occur where the denominator of a rational function is zero, horizontal asymptotes describe the behavior of the function as x approaches infinity, and oblique (or slant) asymptotes appear when the degree of the numerator is one greater than that of the denominator.
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Finding Asymptotes
To find vertical asymptotes, set the denominator equal to zero and solve for x. For horizontal asymptotes, compare the degrees of the numerator and denominator: if they are equal, divide the leading coefficients; if the numerator's degree is less, the asymptote is y=0; if greater, there is no horizontal asymptote. Oblique asymptotes can be found using polynomial long division when the numerator's degree exceeds that of the denominator by one.
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