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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 29
Chapter 4, Problem 29

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

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1
Start by identifying the rational function given: \(f(x) = \frac{x^{2} - 9}{x - 3}\).
Factor the numerator \(x^{2} - 9\) using the difference of squares formula: \(a^{2} - b^{2} = (a - b)(a + b)\), so \(x^{2} - 9 = (x - 3)(x + 3)\).
Rewrite the function using the factored form: \(f(x) = \frac{(x - 3)(x + 3)}{x - 3}\).
Look for common factors in the numerator and denominator. Since \((x - 3)\) appears in both, it can be canceled out, but note that \(x \neq 3\) because it would make the denominator zero.
Determine the vertical asymptotes and holes: the value \(x = 3\) causes the denominator to be zero. Since \((x - 3)\) cancels, there is a hole at \(x = 3\). There are no other values making the denominator zero, so there are no vertical asymptotes.

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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 domains, zeros, and discontinuities, which occur where the denominator equals zero.
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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 and the numerator is nonzero, causing the function to approach infinity or negative infinity. They represent lines the graph approaches but never touches.
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Determining Vertical Asymptotes

Holes in the Graph

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