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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 73b
Chapter 4, Problem 73b

Solve each problem. Work each of the following. Find an equation for a possible corresponding rational function.

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Identify the key characteristics of the rational function you want to construct, such as vertical asymptotes, horizontal or oblique asymptotes, holes, and intercepts. These features will guide the form of the numerator and denominator polynomials.
Determine the vertical asymptotes by setting the denominator equal to zero and solving for the values of \(x\) that make the denominator zero. These values will be the roots of the denominator polynomial.
Decide on the degree and form of the numerator polynomial based on the desired horizontal or oblique asymptote. For example, if the horizontal asymptote is \(y = 0\), the degree of the numerator should be less than the degree of the denominator.
Construct the rational function as a ratio of two polynomials: \(f(x) = \frac{P(x)}{Q(x)}\), where \(Q(x)\) is the denominator polynomial with roots corresponding to vertical asymptotes, and \(P(x)\) is the numerator polynomial chosen to satisfy the horizontal asymptote and any given intercepts.
Verify the function by checking that it has the desired vertical asymptotes, horizontal or oblique asymptotes, and any specified intercepts or holes. Adjust the polynomials if necessary to meet all conditions.

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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), where Q(x) ≠ 0. Understanding the structure of rational functions helps in identifying their behavior, domain restrictions, and possible forms of the equation.
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Domain and Restrictions

The domain of a rational function excludes values that make the denominator zero, as division by zero is undefined. Identifying these restrictions is essential to correctly form the function and understand its graph.
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Constructing Equations from Conditions

To find an equation for a rational function, one must use given conditions such as zeros, asymptotes, or points on the graph. This involves forming polynomials in numerator and denominator that satisfy these conditions.
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