Problem 38 - College Algebra

Question

Find the horisontal asymptote, if there is one, of the graph of each rational function. $$f(x)=\frac{15 x}{3 x^{2}+1}$$

Step-by-Step Solution

Answer

The horizontal asymptote of the rational function \(f(x) = \frac{15x}{3x^{2}+1}\) is at the x-axis (y = 0).

Step 1: Identify Degree of Polynomials

First, identify the degree of the polynomials in the numerator and denominator. The numerator is a polynomial of degree 1, represented as \(15x^{1}\), while the denominator is a polynomial of degree 2, represented as \(3x^{2}\).

Step 2: Compare Polynomial Degrees

Next, compare the degrees of the polynomials. Since the degree of the polynomial in the denominator (2), is higher than the degree of the polynomial in the numerator (1), this implies that the x-axis (y = 0) is the horizontal asymptote.

Step 3: Write down the Asymptote

So, for the function \(f(x) = \frac{15x^{1}}{3x^{2}+1}\), the horizontal asymptote is at the equation \(y = 0\)

Key Concepts

Rational Functions

A rational function is a fraction in which both the numerator and the denominator are polynomials. These functions are important in mathematics as they can model many real-world situations. Understanding how to find the horizontal asymptote of a rational function is particularly useful because it tells us how the function behaves as the input values become very large or very small.

When analyzing our given function, \(f(x)=\frac{15 x}{3 x^{2}+1}\), the numerator, \(15x\), represents a linear polynomial, while the denominator, \(3x^{2}+1\), is a quadratic polynomial. The degrees of these polynomials are crucial in determining the asymptotic behavior of the function. This brings us to our next concept, the degree of a polynomial.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial's expression. It is a simple yet vital concept in algebra, helping to predict the behavior of polynomial expressions under various operations, including division as in the case of rational functions.

In our example, the degree of the numerator, \(15x\), is 1 because the highest power of \(x\) is 1. The denominator \(3x^{2}+1\) has a degree of 2, signaled by the highest power of \(x\), which is 2. Comparing these degrees directly impacts the horizontal asymptote of the rational function. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote will be the x-axis, or \(y=0\).

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as it moves towards infinity or negative infinity. It's a fundamental concept in understanding the big picture of a function's graph. Horizontal asymptotes provide a visual representation of this behavior by showing a line that the function approaches but never actually reaches.

In the case of our rational function, the degrees of the polynomial indicate that as \(x\) approaches infinity, the value of the function approaches 0. That is why the horizontal asymptote is found to be \(y = 0\). The function's values get closer and closer to 0, but will not cross the asymptote, painting a clear picture of the function's behavior at extreme values of \(x\).