Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

5. Rational Functions

Introduction to Rational Functions

College Algebra

5. Rational Functions

Introduction to Rational Functions

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1:43 minutes

Problem 37

Textbook Question

In Exercises 37–44, find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=12x/(3x^2+1)

Verified step by step guidance

Identify the rational function: \( f(x) = \frac{12x}{3x^2 + 1} \).

Determine the degrees of the numerator and the denominator. The degree of the numerator is 1, and the degree of the denominator is 2.

Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).

Recall that for rational functions, if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is always \( y = 0 \).

Conclude that the horizontal asymptote of the function \( f(x) = \frac{12x}{3x^2 + 1} \) is \( y = 0 \).

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Key Concepts

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

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given function f(x) = 12x/(3x^2 + 1), the numerator is a polynomial of degree 1, and the denominator is a polynomial of degree 2. Understanding the structure of rational functions is essential for analyzing their behavior, particularly as x approaches infinity.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the input approaches infinity or negative infinity. For rational functions, the horizontal asymptote can be determined by comparing the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0; if they are equal, it is the ratio of their leading coefficients.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. In the context of rational functions, the degrees of the numerator and denominator are crucial for determining the horizontal asymptote. For the function f(x) = 12x/(3x^2 + 1), the degree of the numerator is 1 and the degree of the denominator is 2, which influences the asymptotic behavior of the function.