Skip to main content
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 35
Chapter 4, Problem 35

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. ƒ(x)=(x2+3x+4)/(x-5)

Verified step by step guidance
1
Identify the given rational function: \(f(x) = \frac{x^2 + 3x + 4}{x - 5}\).
Determine the domain by finding values of \(x\) that make the denominator zero. Set the denominator equal to zero: \(x - 5 = 0\).
Solve for \(x\) to find the excluded value from the domain: \(x = 5\). So, the domain is all real numbers except \(x = 5\).
Analyze the behavior of the function for asymptotes. Since the degree of the numerator (2) is greater than the degree of the denominator (1), there is an oblique (slant) asymptote.
Find the oblique asymptote by performing polynomial long division of \(x^2 + 3x + 4\) by \(x - 5\) to express \(f(x)\) as a polynomial plus a remainder over \(x - 5\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

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 helps analyze its behavior, including domain restrictions and asymptotes.
Recommended video:
6:04
Intro to Rational Functions

Domain of a Rational Function

The domain consists of all real numbers except where the denominator equals zero, as division by zero is undefined. For f(x) = (x² + 3x + 4)/(x - 5), x ≠ 5 is excluded from the domain.
Recommended video:
6:04
Intro to Rational Functions

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero, and horizontal or oblique asymptotes depend on the degrees of numerator and denominator polynomials.
Recommended video:
6:24
Introduction to Asymptotes
Related Practice
Textbook Question

Solve each problem. Is x+1 a factor of ƒ(x)=x3+2x2+3x+2?

Textbook Question

Solve each problem. Use Descartes' rule of signs to determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of ƒ(x)=x3+3x2−4x−2ƒ(x)=x^3+3x^2-4x-2.

Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=(3x-1)(x+2)2

Textbook Question

Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. ƒ(x) = -3x2 + 24x - 46

Textbook Question

For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = 2x2 - 3x-3; k = 2

2
views
Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=x2(x-5)(x+3)(x-1)