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Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 3 - Polynomial and Rational FunctionsProblem 70
Chapter 4, Problem 70

Graph each rational function. See Examples 5–9.
ƒ(x)=x/(4−x2)ƒ(x)=x/(4-x^2)

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1
Identify the rational function given: \(f(x) = \frac{x}{4 - x^2}\).
Determine the domain by finding values of \(x\) that make the denominator zero. Solve \(4 - x^2 = 0\) to find vertical asymptotes.
Find the vertical asymptotes by setting the denominator equal to zero and solving for \(x\): \(4 - x^2 = 0\) implies \(x^2 = 4\), so \(x = \pm 2\).
Find the horizontal asymptote by analyzing the degrees of the numerator and denominator. Since the degree of the denominator (2) is greater than the numerator (1), the horizontal asymptote is \(y = 0\).
Find the intercepts: For the \(y\)-intercept, evaluate \(f(0)\); for the \(x\)-intercept, set the numerator equal to zero and solve for \(x\). Then, plot these points along with the asymptotes to sketch the graph.

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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 its domain, zeros, and behavior depends on analyzing both numerator and denominator. For example, f(x) = x/(4 - x^2) involves a quadratic denominator that affects the function's properties.
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Intro to Rational Functions

Domain and Vertical Asymptotes

The domain of a rational function excludes values that make the denominator zero. These values often correspond to vertical asymptotes, where the function approaches infinity or negative infinity. For f(x) = x/(4 - x^2), setting 4 - x^2 = 0 finds vertical asymptotes at x = ±2.
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Determining Vertical Asymptotes

Graphing Rational Functions and Asymptotes

Graphing involves plotting intercepts, identifying asymptotes, and analyzing end behavior. Horizontal or oblique asymptotes describe the function's behavior as x approaches infinity. For f(x) = x/(4 - x^2), the horizontal asymptote is y = 0, since the degree of the denominator is higher than the numerator.
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Related Practice
Textbook Question

Find a polynomial function f of least degree having the graph shown. (Hint: See the NOTE following Example 4.)

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Textbook Question

Solve each rational inequality. Give the solution set in interval notation. 1 /{x2 - 4x + 3} ≤ 1 /{ 3 - x}

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Textbook Question

The remainder theorem indicates that when a polynomial ƒ(x) is divided by x-k, the remainder is equal to ƒ(k). Consider the polynomial function ƒ(x) = x3 - 2x2 - x+2. Use the remainder theorem to find each of the following. Then determine the coordinates of the corresponding point on the graph of ƒ(x). ƒ (1)

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Textbook Question

Graph each rational function. See Examples 5–9.

ƒ(x)=[(x+6)(x−2)]/[(x+3)(x−4)]ƒ(x)=[(x+6)(x-2)]/[(x+3)(x-4)]

Textbook Question

Graph each rational function. ƒ(x)=[(x+3)(x-5)]/[(x+1)(x-4)]

Textbook Question

Graph each rational function. ƒ(x)=5x/(x2-1)