Textbook Question
Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. (x+1)(x+2)(x+3)≥0
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Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
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)
Verified step by step guidance1
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.
Recommended video:
6:04
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.
Recommended video:
3:12Determining 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.
Recommended video:
3:34Determining Removable Discontinuities (Holes)
Related Practice
Textbook Question
Divide using synthetic division. (2x5−3x4+x3−x2+2x−1)/(x+2)
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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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Textbook Question
In Exercises 27–29, divide using long division.
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Textbook Question
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)=x2+3x−10
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Textbook Question
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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