Problem 59 - College Algebra and Calculus: An Applied Approach

Question

Write a rational function \(f\) that has the specified characteristics. (There are many correct answers.) Vertical asymptotes: \(x=-2, x=1\) Horizontal asymptote: None

Step-by-Step Solution

Answer

The rational function is \(f(x) = x^2 / (x + 2)(x - 1)\)

Step 1: Formulate the function

Firstly, establish the expression for the function. The vertical asymptotes occur where the denominator of the rational function equals to zero. In this case, \(x = -2\) and \(x = 1\) are the vertical asymptotes, then these values must make the denominator equal to zero. Therefore, the denominator can be expressed as \((x + 2)(x - 1)\). For the condition of no horizontal asymptote, a polynomial of degree 2 (one degree higher than that of the denominator) can be chosen for the numerator. Hence the rational function could be \(f(x) = x^2 / (x + 2)(x - 1)\)

Step 2: Finalize the rational function

The rational function that satisfies the provided conditions is \(f(x) = x^2 / (x + 2)(x - 1)\)

Key Concepts

Vertical Asymptotes

In a rational function, vertical asymptotes are the values of \(x\) where the function is undefined. This happens because the denominator of the function becomes zero at these points, causing the function to 'blow up' towards infinity. It is a characteristic sharp feature of rational functions.
Understanding where vertical asymptotes occur is crucial for graphing rational functions and analyzing their behavior.

• Vertical asymptotes are found by setting the denominator equal to zero and solving for \(x\).
• If a rational function has a factor in the denominator that is not canceled out by the numerator, then a vertical asymptote occurs where that factor is zero.

For example, if we consider the function with vertical asymptotes at \(x = -2\) and \(x = 1\), we set \((x + 2)(x - 1) = 0\) and solve for \(x\), which confirms the points where the vertical asymptotes occur.

Denominator

The denominator of a rational function plays a key role in shaping the behavior of the function. It determines where the vertical asymptotes will occur, as well as where the function is undefined.
To analyze a rational function, you need to understand the polynomial in the denominator.

• The roots of the denominator are the \(x\) values that cause the function to be zero, leading to vertical asymptotes.
• A well-crafted denominator helps you predict the graph and characteristics of the rational function.

In the given function example, the denominator \((x + 2)(x - 1)\) indicates that the expression is undefined at \(x = -2\) and \(x = 1\). It is a polynomial expression determining the rational function's backbone.

Polynomial Degree

The degree of the polynomial in both the numerator and the denominator influences the overall behavior of a rational function, including the asymptotic behavior where horizontal asymptotes may or may not appear.
For our problem, where no horizontal asymptote is required, we strategically ensure that the numerator's polynomial degree is higher than that of the denominator.

• A rational function will not have a horizontal asymptote if the degree of the numerator exceeds that of the denominator. This means the degree of the numerator should be at least one greater.
• This difference in degrees dictates how the function stretches and grows towards very large \(|x|\) values, heading towards infinity rather than leveling out.

For example, choosing a numerator like \(x^2\), which is a quadratic expression (degree 2), while keeping the denominator as a product of linear factors (degree 1), results in no horizontal asymptote.