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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 51
Chapter 4, Problem 51

Work each problem. Which function has a graph that does not have a vertical asymptote?
A. ƒ(x)=1/(x2+2)
B. ƒ(x)=1/(x2-2)
C. ƒ(x)=3/x2
D. ƒ(x)=(2x+1)/(x-8)

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1
Recall that vertical asymptotes occur where the denominator of a rational function is equal to zero, causing the function to be undefined at those points.
For each function, identify the denominator and set it equal to zero to find potential vertical asymptotes.
For option A: The denominator is \(x^2 + 2\). Set \(x^2 + 2 = 0\) and solve for \(x\).
For option B: The denominator is \(x^2 - 2\). Set \(x^2 - 2 = 0\) and solve for \(x\).
For option C: The denominator is \(x^2\). Set \(x^2 = 0\) and solve for \(x\). For option D: The denominator is \(x - 8\). Set \(x - 8 = 0\) and solve for \(x\). Then determine which function's denominator has no real solutions, indicating no vertical asymptotes.

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Key Concepts

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

Vertical Asymptotes

Vertical asymptotes occur in the graph of a function where the function approaches infinity or negative infinity as the input approaches a specific value, typically where the denominator of a rational function is zero and the numerator is nonzero.
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Determining Vertical Asymptotes

Domain of Rational Functions

The domain of a rational function includes all real numbers except where the denominator equals zero, since division by zero is undefined. Identifying these values helps determine where vertical asymptotes may exist.
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Intro to Rational Functions

Analyzing Quadratic Expressions in Denominators

Quadratic expressions in denominators can be factored or analyzed using the discriminant to find real roots. If the quadratic has no real roots, the denominator never equals zero, meaning no vertical asymptotes occur.
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Rationalizing Denominators
Related Practice
Textbook Question

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=(x2+x−2)5(x−1+3)2ƒ(x)=(x^2+x-2)^5(x-1+\(\sqrt\)3)^2

Textbook Question

Use synthetic division to determine whether the given number k is a zero of the polynomial function. If it is not, give the value of ƒ(k). ƒ(x) = 2x3 - 6x2 -9x + 4; k=1

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Textbook Question

Graph each polynomial function. ƒ(x)=-2x4+7x3-4x2-4x

Textbook Question

Connecting Graphs with Equations Find a quadratic function f having the graph shown. (Hint: See the Note following Example 3.)

Textbook Question

Work each problem. Which function has a graph that does not have a horizontal asymptote?

A. ƒ(x)=(2x-7)/(x+3)

B. ƒ(x)=3x/(x2-9)

C. ƒ(x)=(x2-9)/(x+3)

D. ƒ(x)=(x+5)/(x+2)(x-3)

Textbook Question

Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers given. ƒ(x)=2x4-4x2+4x-8; 1 and 2