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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 41
Chapter 4, Problem 41

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(x2-1)/(x+3)

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Identify the rational function given: \(f(x) = \frac{x^2 - 1}{x + 3}\).
Find the vertical asymptotes by setting the denominator equal to zero and solving for \(x\): solve \(x + 3 = 0\) to find values where the function is undefined.
Determine the horizontal or oblique asymptotes by comparing the degrees of the numerator and denominator polynomials. Here, the numerator degree is 2 and the denominator degree is 1.
Since the numerator degree is greater than the denominator degree by exactly 1, perform polynomial long division of \(x^2 - 1\) by \(x + 3\) to find the oblique asymptote.
Write the oblique asymptote as the quotient from the division (ignoring the remainder), and summarize the vertical asymptote(s) found in step 2.

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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 where the denominator of a rational function equals zero, causing the function to approach infinity or negative infinity. To find them, set the denominator equal to zero and solve for x, ensuring these values do not cancel with factors in the numerator.
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Determining Vertical Asymptotes

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches infinity or negative infinity. For rational functions, compare the degrees of the numerator and denominator: if the numerator's degree is less, the asymptote is y=0; if equal, it's the ratio of leading coefficients.
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Determining Horizontal Asymptotes

Oblique (Slant) Asymptotes

Oblique asymptotes occur when the degree of the numerator is exactly one more than the denominator. They are found by performing polynomial division of numerator by denominator; the quotient (without the remainder) gives the equation of the slant asymptote.
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Introduction to Asymptotes
Related Practice
Textbook Question

For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = x2 + 4; k = 2i

Textbook Question

Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. ƒ(x) = x2 - 4x + 3

Textbook Question

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(4-3x)/(2x+1)

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4.

ƒ(x)=−3x4−5x3+2x2ƒ(x)=-3x^4-5x^3+2x^2

Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=2x3(x2-4)(x-1)

Textbook Question

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(2x+6)/(x-4)