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) = -(x - 2)2 - 5
2
views
Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
Not the one you use?Change textbookSelect a different textbook
Table of contents
Chapter 4, Problem 39
Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(4-3x)/(2x+1)
Verified step by step guidance1
Identify the rational function given: \(f(x) = \frac{4 - 3x}{2x + 1}\).
Find the vertical asymptotes by setting the denominator equal to zero and solving for \(x\): \(2x + 1 = 0\). Solve this equation to find the value(s) of \(x\) where the function is undefined.
Determine the horizontal or oblique asymptotes by comparing the degrees of the numerator and denominator polynomials. The numerator degree is the highest power of \(x\) in \(4 - 3x\), and the denominator degree is the highest power of \(x\) in \(2x + 1\).
Since the degrees of numerator and denominator are equal (both degree 1), find the horizontal asymptote by dividing the leading coefficients of the numerator and denominator. The horizontal asymptote is \(y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}\).
If the degree of the numerator were exactly one more than the degree of the denominator, you would perform polynomial long division to find the oblique asymptote. In this case, since degrees are equal, no oblique asymptote exists.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5mWas 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). Understanding the behavior of rational functions, especially where the denominator is zero, is essential for identifying asymptotes and graphing the function.
Recommended video:
6:04
Intro to Rational Functions
Vertical Asymptotes
Vertical asymptotes occur where the denominator of a rational function equals zero and the numerator is nonzero, causing the function to approach infinity or negative infinity. These lines are vertical and indicate values of x that the function cannot take.
Recommended video:
3:12Determining Vertical Asymptotes
Horizontal and Oblique Asymptotes
Horizontal asymptotes describe the end behavior of a function as x approaches infinity or negative infinity, determined by comparing degrees of numerator and denominator. Oblique (slant) asymptotes occur when the numerator's degree is exactly one more than the denominator's, found via polynomial division.
Recommended video:
4:48Determining Horizontal Asymptotes
Related Practice
Textbook Question
Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4.
Textbook Question
Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(x2-1)/(x+3)
Textbook Question
Simple Interest Simple interest varies jointly as principal and time. If \$1000 invested for 2 yr earned \$70, find the amount of interest earned by \$5000 invested for 5 yr.
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)
Textbook Question
Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=-x3+x2+2x