Problem 34 - College Algebra

Question

Find the horizontal asymptote, if any, of the graph of each rational function. $$h(x)=\frac{15 x^{3}}{3 x^{2}+1}$$

Step-by-Step Solution

Answer

The given function does not have a horizontal asymptote.

Step 1: Finding the degree of the polynomial

The polynomial in the numerator is \(15x^3\) and the one in the denominator is \(3x^2+1\). Their degrees are respectively 3 and 2. The degree of a polynomial is the highest power of x in that polynomial.

Step 2: Comparing the degrees

Compare the degrees of the polynomials in the numerator and the denominator. If the degree of the numerator is greater than the degree of the denominator, as in this case, then the graph of the function will not have any horizontal asymptote.

Step 3: Concluding the absence of the horizontal asymptote

Since the degree of the numerator is greater than the degree of the denominator, this function does not have a horizontal asymptote. In general, if the degree of the numerator is greater than the degree of the denominator, the rational function does not have any horizontal asymptote.

Key Concepts

Rational Function

A rational function is a type of function you often encounter in algebra—it's formed when you divide one polynomial by another. Think of it as a fraction where both the numerator and the denominator are polynomials. We represent rational functions as \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) \) is not equal to zero.

For the given exercise, the function \( h(x) = \frac{15x^3}{3x^2+1} \) is a rational function. Here, \( 15x^3 \) represents the polynomial in the numerator, and \( 3x^2+1 \) is the polynomial in the denominator. In algebra, rational functions are analyzed to understand their properties, such as intercepts, domain, range, and especially their asymptotic behavior.

Polynomial Degree

The degree of a polynomial is a fundamental concept that defines the highest power of the variable \( x \) present in the polynomial. For instance, in the polynomial \( a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \) with non-zero coefficient \( a_n \) and \( n \) as a non-negative integer, the degree is \( n \).

In our exercise, \( h(x) \) is presented in terms of two polynomials, where the degree of the numerator polynomial \( 15x^3 \) is 3, and the degree of the denominator polynomial \( 3x^2+1 \) is 2. The relationship between the degrees of these polynomials plays a pivotal role in determining the presence or absence of horizontal asymptotes.

Asymptotic Behavior

Asymptotic behavior describes how a function operates as it moves towards a specific value or infinity. It is a way to understand the trends and directions of a function's graph—where it's heading, but never quite reaches. This concept is particularly useful when dealing with rational functions.

A horizontal asymptote is a horizontal line that a function approaches as \( x \) moves towards infinity or negative infinity. It gives us a “behavioral boundary” for the function's graph. In simpler terms, it's like a rule that tells the graph how to behave far out to the left or right - the ultimate path the graph tends to follow, though it might never actually touch the line.

In the context of our exercise, we're looking at the rational function \( h(x) = \frac{15x^3}{3x^2+1} \). The degrees of the numerator and the denominator polynomials determine the horizontal asymptote. Since the degree of the numerator is higher, we find that this particular function does not level off to a horizontal line at any value, which means it lacks a horizontal asymptote. Thus, its asymptotic behavior is such that as \( x \) becomes very large or very small, the values of \( h(x) \) increase without bound and do not approach a constant value.