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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblema 33
Capitolo 4, Problema 33

Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. h(x)=(x+7)/(x2+4x−21)

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Start by identifying the denominator of the rational function, which is \(x^{2} + 4x - 21\).
Factor the denominator to find its roots. To factor \(x^{2} + 4x - 21\), look for two numbers that multiply to \(-21\) and add to \(4\).
Write the denominator as a product of two binomials based on the factors found: \(x^{2} + 4x - 21 = (x + a)(x + b)\), where \(a\) and \(b\) are the numbers from the previous step.
Set each factor equal to zero to find the values of \(x\) that make the denominator zero: \(x + a = 0\) and \(x + b = 0\). These values are potential vertical asymptotes or holes.
Check if any of these \(x\)-values also make the numerator zero. If a value makes both numerator and denominator zero, it corresponds to a hole; if it only makes the denominator zero, it corresponds to a vertical asymptote.

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Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding the behavior of rational functions involves analyzing their numerators and denominators, especially where the denominator equals zero, which affects the domain and graph.
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Intro to Rational Functions

Vertical Asymptotes

Vertical asymptotes occur at values of x where the denominator of a rational function is zero but the numerator is not zero, causing the function to approach infinity or negative infinity. Identifying these points helps describe the function's behavior near undefined values.
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Determining Vertical Asymptotes

Holes in the Graph

Holes occur when a factor cancels out from both numerator and denominator, resulting in a removable discontinuity. At these x-values, the function is undefined, but the limit exists, indicating a 'hole' rather than an asymptote on the graph.
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Determining Removable Discontinuities (Holes)
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