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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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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)
Related Practice
Textbook Question

Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=4; -2, 5, and 3+2i are zeros; f(1) = -96

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Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=2x−x2+3

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