Problem 43 - College Algebra

Question

Find the horisontal asymptote, if there is one, of the graph of each rational function. $$f(x)=\frac{-2 x+1}{3 x+5}$$

Step-by-Step Solution

Answer

The horizontal asymptote of the function \(f(x) = \frac{-2x+1}{3x+5}\) is \(y = -2/3\).

Step 1: Identify the Degrees of the Polynomials

Identify the degree of the numerator and the degree of the denominator. In our function \(f(x) = \frac{-2x+1}{3x+5}\), we can see that the degree of the numerator (which is the highest power of x) is 1 and the degree of the denominator is also 1.

Step 2: Determine the Asymptote Based on the Degrees

When the degrees of both the numerator and the denominator are equal (which is the case here, as both are 1), the horizontal asymptote is the ratio of the leading coefficients of the numerator and the denominator. The leading coefficient is the coefficient of the highest degree term. In our case, the leading coefficient of the numerator is -2 and that of the denominator is 3.

Step 3: Formulate the Asymptote

The ratio of the leading coefficients of the numerator to the denominator is \(-2/3\). Thus, the equation of the horizontal asymptote is \(y = -2/3\).

Key Concepts

Rational Functions

Rational functions are an integral part of algebra that appear frequently in both mathematics and real-world applications. These functions are expressed in the form \(f(x) = \frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomials. The key characteristic of rational functions is that the denominator \(q(x)\) must not be equal to zero, as division by zero is undefined.

One of the features of rational functions is that they can produce horizontal, vertical, or oblique asymptotes. These asymptotes help describe the behavior of the function as \(x\) approaches specific values, such as infinity or points not included in the domain. By examining asymptotes, we can better understand the function's graph and predict its behavior over its domain.

Polynomials

A polynomial is an algebraic expression made up of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, the polynomial \(-2x + 1\) features the variable \(x\) with a coefficient of \(-2\), and a constant term of 1.

Polynomials can be classified based on their degrees, leading coefficients, or the number of terms. Understanding these classifications helps not only in simplifying expressions but also in solving polynomial equations and analyzing rational functions. When polynomials are used in the numerator or denominator of a fraction, we obtain a rational function, like the example function \(f(x) = \frac{-2x+1}{3x+5}\).

Leading Coefficients

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. In a rational function, the leading coefficients of both the numerator and the denominator play a crucial role in determining the horizontal asymptote.

For instance, in \(f(x) = \frac{-2x+1}{3x+5}\), the leading coefficient of the numerator is \(-2\) (from the term \(-2x\)), while the leading coefficient of the denominator is \(3\) (from the term \(3x\)).

When the degrees of both the numerator and denominator are equal, the horizontal asymptote is simply the fraction of these leading coefficients, as noted in the solution steps. This ratio, \(y = \frac{-2}{3}\), determines how the graph of the rational function behaves at extreme values of \(x\).

Degree of Polynomials

The degree of a polynomial is determined by the highest power of the variable in the expression. Understanding the degree of polynomials is essential when analyzing rational functions, particularly when identifying asymptotic behavior.

Polynomial degrees are used to determine which terms dominate the function as the variable gets very large or very small. In \(f(x) = \frac{-2x+1}{3x+5}\), both the numerator and the denominator have a degree of 1, because the highest power of \(x\) is 1 in both polynomials.

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).
• If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
• If the degree of the numerator is greater, there is no horizontal asymptote.

These rules allow us to quickly understand the end behavior of rational functions.