Textbook Question
Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.
As __
1315
views
5. Rational Functions
Introduction to Rational Functions
1:43 minutesProblem 37
Textbook Question
Find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=12x/(3x2+1)
Verified step by step guidance1
Identify the degrees of the numerator and denominator polynomials in the rational function \(f(x) = \frac{12x}{3x^{2} + 1}\).
Recall that the degree of the numerator is 1 (since \$12x\( is a first-degree polynomial) and the degree of the denominator is 2 (since \)3x^{2}$ is a second-degree polynomial).
Use the rule for horizontal asymptotes of rational functions: if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).
Since the degree of the numerator (1) is less than the degree of the denominator (2), conclude that the horizontal asymptote is \(y = 0\).
Write the final conclusion that the graph of \(f(x)\) has a horizontal asymptote at \(y = 0\).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mKey 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 involves analyzing the degrees and coefficients of the numerator and denominator polynomials.
Recommended video:
6:04
Intro to Rational Functions
Horizontal Asymptotes
A horizontal asymptote describes the behavior of a function as x approaches infinity or negative infinity. It is a horizontal line y = L that the graph approaches but does not necessarily touch, indicating the end behavior of the function.
Recommended video:
4:48Determining Horizontal Asymptotes
Degree Comparison of Polynomials
To find horizontal asymptotes of rational functions, compare the degrees of the numerator and denominator polynomials. If the numerator's degree is less, the asymptote is y=0; if equal, it is the ratio of leading coefficients; if greater, there is no horizontal asymptote.
Recommended video:
Guided course
05:16Standard Form of Polynomials
Related Videos
Related Practice