Problem 271 - Calculus Volume 1

Question

For the following exercises, find the horizontal and vertical asymptotes. $$ f(x)=x-\frac{9}{x} $$

Step-by-Step Solution

Answer

Horizontal asymptote: None. Vertical asymptote: \(x=0\).

Step 1: Identify Horizontal Asymptotes

To find the horizontal asymptotes, compare the degrees of the polynomial terms in the numerator and the denominator. In the function \( f(x) = x - \frac{9}{x} \), this can be rewritten as \( f(x) = \frac{x^2 - 9}{x} \). The degree of the numerator is 2, and the degree of the denominator is 1. Since the degree of the numerator is greater than the degree of the denominator, there are no horizontal asymptotes.

Step 2: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function equals zero and the numerator does not simultaneously equal zero. In the function \( f(x) = x - \frac{9}{x} = \frac{x^2 - 9}{x} \), set the denominator \( x \) to zero. Solving \( x = 0 \) indicates a potential vertical asymptote at \( x = 0 \). Since division by zero is undefined and the expression does not simplify further or resolve \( x = 0 \), \( x = 0 \) is indeed a vertical asymptote.

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes help us understand the long-term behavior of a function as the input value becomes very large or very small. For rational functions, which are the ratio of two polynomials, the horizontal asymptote is determined by comparing the degrees of the polynomial in the numerator and the polynomial in the denominator.

When the degree of the numerator is less than that of the denominator, the horizontal asymptote is the x-axis, or the line \( y = 0 \). When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator. When the degree of the numerator is greater than the degree of the denominator, as in our example function \( f(x) = \frac{x^2 - 9}{x} \), there is no horizontal asymptote.

Knowing how to identify horizontal asymptotes allows us to predict how the function behaves at the ends and is an essential skill when graphing rational functions.

Vertical Asymptotes

Vertical asymptotes represent the points where a function's value approaches infinity as the input values approach a certain point. They occur in rational functions when the denominator equals zero, causing the function to be undefined at that point. However, for a vertical asymptote to exist, the numerator must not be zero simultaneously at that point.

In our example function, \( f(x) = \frac{x^2 - 9}{x} \), we set the denominator equal to zero to find potential vertical asymptotes. Solving \( x = 0 \) gives us such a point, indicating a vertical asymptote at \( x = 0 \). This is because the function becomes undefined as it involves division by zero, and the numerator \( x^2 - 9 \) does not cross zero simultaneously.

Understanding vertical asymptotes is crucial in the analysis of rational functions as it helps in identifying the function's discontinuities and avoiding undefined points during graphing.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. In rational functions, which are divisions of two polynomials, understanding their degrees helps in determining the asymptotic behavior of the function.

For the given function \( f(x) = x - \frac{9}{x} \), it can be rewritten as \( f(x) = \frac{x^2 - 9}{x} \). Here, the degree of the polynomial in the numerator is 2, and the degree in the denominator is 1. The difference in degrees influences the type of asymptote the function has. As shown, since the numerator's degree is higher, there is no horizontal asymptote, although a vertical asymptote exists at \( x = 0 \).

Grasping the concept of polynomial degree equips students with the ability to assess function behavior quickly and efficiently, especially when dealing with rational functions.