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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 61
Chapter 4, Problem 61

Follow the seven steps to graph each rational function. f(x)=2x2/(x2−1)

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Identify the domain of the function by finding the values of \(x\) that make the denominator zero. Solve the equation \(x^{2} - 1 = 0\) to find these values, since the function is undefined there.
Find the vertical asymptotes by setting the denominator equal to zero and solving for \(x\). These are the lines where the function approaches infinity or negative infinity.
Determine the horizontal or oblique asymptote by comparing the degrees of the numerator and denominator. Since both numerator and denominator are degree 2, find the horizontal asymptote by dividing the leading coefficients.
Find the intercepts: For the \(y\)-intercept, evaluate \(f(0)\). For the \(x\)-intercepts, set the numerator equal to zero and solve for \(x\).
Analyze the behavior of the function near the vertical asymptotes and at points in each interval determined by the vertical asymptotes to understand how the graph behaves on those intervals.

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 its domain, where the denominator Q(x) ≠ 0, is essential to avoid undefined points and to analyze behavior such as vertical asymptotes.
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Intro to Rational Functions

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero, and horizontal or oblique asymptotes describe end behavior as x approaches infinity or negative infinity.
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Introduction to Asymptotes

Graphing Steps for Rational Functions

Graphing involves identifying domain restrictions, intercepts, asymptotes, and behavior near asymptotes, then plotting points to sketch the curve. Following a systematic seven-step process ensures a complete and accurate graph.
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How to Graph Rational Functions
Related Practice
Textbook Question

Find the inverse of f(x) = x3 + 2

Textbook Question

Follow the seven steps to graph each rational function. f(x)=−x/(x+1)

Textbook Question

Find the domain of each function. f(x)=2x2−5x+2f(x) = \(\sqrt{2x^2 - 5x + 2}\)

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Textbook Question

Among all pairs of numbers whose sum is 16, find a pair whose product is as large as possible. What is the maximum product?

Textbook Question

Find the domain of each function. f(x)=2xx+1−1f(x) = \(\sqrt{\frac{2x}{x + 1}\) - 1}

Textbook Question

Find the domain of each function. f(x)=14x2−9x+2f(x) = \(\frac{1}{\sqrt{4x^2 - 9x + 2}\)}