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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 31
Chapter 4, Problem 31

Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. g(x)=(x−3)/(x2−9)

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1
Start by identifying the rational function given: \(g(x) = \frac{x - 3}{x^{2} - 9}\).
Factor the denominator to find values that make it zero: \(x^{2} - 9\) factors as \((x - 3)(x + 3)\).
Set the denominator equal to zero to find potential vertical asymptotes or holes: solve \((x - 3)(x + 3) = 0\), which gives \(x = 3\) and \(x = -3\).
Check if any factor in the numerator cancels with a factor in the denominator. Since the numerator is \(x - 3\), it cancels with the \((x - 3)\) factor in the denominator, indicating a hole at \(x = 3\).
The remaining factor in the denominator, \((x + 3)\), does not cancel, so \(x = -3\) is a vertical asymptote.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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