Rational Functions

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What is a rational function?

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A rational function is a function that can be expressed as the quotient of two polynomials, i.e., \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials.

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What are vertical asymptotes?

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Vertical asymptotes are lines \( x = a \) where a rational function approaches infinity or negative infinity as the input approaches \( a \). They occur where the denominator is zero and the numerator is not zero.

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What are horizontal asymptotes?

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Horizontal asymptotes are lines \( y = b \) that a rational function approaches as \( x \) approaches infinity or negative infinity. They indicate the end behavior of the function.

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How do you find vertical asymptotes of a rational function?

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To find vertical asymptotes, set the denominator equal to zero and solve for \( x \). The values of \( x \) that make the denominator zero are the vertical asymptotes.

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How do you find horizontal asymptotes of a rational function?

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To find horizontal asymptotes, compare the degrees of the numerator and denominator: 1) If degree of numerator < degree of denominator, \( y = 0 \). 2) If degree of numerator = degree of denominator, \( y = \frac{leading coefficient of P}{leading coefficient of Q} \). 3) If degree of numerator > degree of denominator, there is no horizontal asymptote.

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What is an extraneous solution?

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An extraneous solution is a solution that emerges from the process of solving an equation but does not satisfy the original equation.

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x = 11.

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x = -7, 4.

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What does it mean to check for extraneous solutions?

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Checking for extraneous solutions means substituting the found solutions back into the original equation to verify if they hold true.

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What is the parent function of a rational function?

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The parent function of a rational function is \( f(x) = \frac{1}{x} \). It serves as the basic form from which other rational functions are derived.