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. The general form is f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials. Understanding the degrees of the polynomials in the numerator and denominator is crucial for analyzing the behavior of the function, particularly in relation to asymptotes.
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Intro to Rational Functions
Asymptotes
Asymptotes are lines that the graph of a function approaches but never touches. There are three types: vertical asymptotes occur where the denominator is zero (and the function is undefined), horizontal asymptotes describe the behavior of the function as x approaches infinity, and oblique (or slant) asymptotes occur when the degree of the numerator is one greater than that of the denominator.
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Introduction to Asymptotes
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, the asymptote is y = leading coefficient of P / leading coefficient of Q; if the degree of the numerator is less, the asymptote is y = 0. For oblique asymptotes, perform polynomial long division when the numerator's degree is one greater than the denominator's.
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Introduction to Asymptotes