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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 72
Chapter 4, Problem 72

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

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Identify the rational function given: \(f(x) = \frac{(x+3)(x-5)}{(x+1)(x-4)}\). This function is a ratio of two polynomials.
Find the domain by determining the values of \(x\) that make the denominator zero. Solve \( (x+1)(x-4) = 0 \) to find the vertical asymptotes.
Find the zeros of the function by setting the numerator equal to zero: \( (x+3)(x-5) = 0 \). These values are the x-intercepts of the graph.
Determine the horizontal asymptote by comparing the degrees of the numerator and denominator. Since both are degree 2, the horizontal asymptote is the ratio of the leading coefficients.
Plot the vertical asymptotes, zeros, and horizontal asymptote on the coordinate plane. Then, choose test points in each interval to find the sign of \(f(x)\) and sketch the graph accordingly.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Functions and Their Domains

A rational function is a ratio of two polynomials. Its domain includes all real numbers except where the denominator equals zero, causing vertical asymptotes or holes. Identifying these restrictions is essential before graphing.
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Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur at values that make the denominator zero, while horizontal or oblique asymptotes describe end behavior based on the degrees of numerator and denominator polynomials.
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Intercepts and Sign Analysis

Finding x-intercepts involves setting the numerator equal to zero, while y-intercepts are found by evaluating the function at x=0. Sign analysis between intercepts and asymptotes helps determine where the function is positive or negative, guiding accurate graphing.
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Related Practice
Textbook Question

Height of an Object If an object is projected upward from an initial height of 100 ft with an initial velocity of 64 ft per sec, then its height in feet after t seconds is given by s(t)=−16t2+64t+100s(t) = -16t^2 + 64t + 100. Find the number of seconds it will take the object to reach its maximum height. What is this maximum height?

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Textbook Question

Find a polynomial function ƒ(x) of least degree having only real coefficients and zeros as given. Assume multiplicity 1 unless otherwise stated. 2-i and 6-3i

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Textbook Question

Solve each problem. Work each of the following. Sketch the graph of a function that does not intersect its horizontal asymptote y=1, has the line x=3 as a vertical asymptote, and has x-intercepts (2, 0) and (4, 0).

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Textbook Question

Graph each rational function. ƒ(x)=(3x2+3x-6)/(x2-x-12)

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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

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)]

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