Problem 56 - Calculus

Question

In Problems \(52-57\), give an example of: A function that has exactly 17 vertical asymptotes.

Step-by-Step Solution

Answer

The function \( f(x) = \frac{1}{(x-1)(x-2)...(x-17)} \) has exactly 17 vertical asymptotes.

Step 1: Understand Vertical Asymptotes

Vertical asymptotes occur in a function when the denominator of a rational function approaches zero while the numerator does not equal zero. Specifically, for a function \( f(x) = \frac{p(x)}{q(x)} \), vertical asymptotes are found where \( q(x) = 0 \). Thus, we need 17 distinct values where \( q(x) \) equals zero.

Step 2: Define a Function with Multiple Vertical Asymptotes

To create a function with 17 vertical asymptotes, design \( q(x) \) to have 17 unique linear factors, each equating to zero at a different value of \( x \). For instance, use \( q(x) = (x-a_1)(x-a_2)...(x-a_{17}) \), where each \( a_i \) is a distinct constant.

Step 3: Formulate the Example Function

Construct the function as follows: \( f(x) = \frac{1}{(x-1)(x-2)(x-3)...(x-17)} \). Each factor \( x-a_i \) contributes a vertical asymptote at \( x = a_i \).

Step 4: Verify Vertical Asymptote Count

Verify that substituting each \( x = a_i \) individually into \( f(x) \) makes the denominator zero, confirming 17 distinct vertical asymptotes. None of these points should cancel with factors in the numerator which in this case is \(1\).

Key Concepts

Rational Functions

Rational functions are a fascinating topic within algebra. They are formed by dividing two polynomials, much like a fraction with a numerator and a denominator. In mathematical terms, you might see a rational function expressed as \( f(x) = \frac{p(x)}{q(x)} \). Here, \( p(x) \) and \( q(x) \) are polynomial functions.

Since rational functions are essentially fractions, they exhibit behaviors found with fractions, such as division by zero issues. These issues manifest as vertical asymptotes when evaluating the domain of the function. Understanding rational functions deeply involves exploring these behaviors.

• The numerator \( p(x) \) affects the function's overall zeroes when set equal to zero.
• The denominator \( q(x) \) is crucial in determining vertical asymptotes since setting it to zero reveals where the function is undefined.

Grasping these concepts helps in dealing with and graphing rational functions, unveiling more about their structure and characteristics.

Polynomial Denominators

A polynomial denominator is the heart of analyzing rational functions when it comes to vertical asymptotes. It comes in the form of a polynomial equation in the denominator of a rational function. For vertical asymptotes, our main concern is the values of \( x \) that make the denominator zero.

A polynomial can often be broken down into factors, which could indicate potential vertical asymptotes. Imagine a polynomial like \( q(x) = (x - a_1)(x - a_2)...(x - a_n) \). Each distinct factor \( x - a_i \) contributes to a potential vertical asymptote when equal to zero.

For example, if we have a polynomial denominator as highlighted in the original exercise, such as \( q(x) = (x-1)(x-2)...(x-17) \), it has been designed to produce 17 distinct zeroes leading to 17 vertical asymptotes.

• The degree of the polynomial in the denominator determines how many potential vertical asymptotes might exist.
• It's crucial that the corresponding numerator does not cancel out these factors, ensuring the presence of true vertical asymptotes.

This consideration is critical for both theoretical and practical applications of functions in mathematical modeling.

Asymptotic Behavior

Asymptotic behavior in rational functions is an essential concept for understanding how the function behaves as \( x \) approaches specific values or infinity. There are different types of asymptotes, but when discussing vertical asymptotes, we're specifically interested in \( x \) values where the function grows unbounded, typically at points where the denominator reaches zero.

To identify vertical asymptotes, one can inspect the values of \( x \) that make the denominator zero, while ensuring these values don't cause corresponding numerator factors to zero out. This zeroing out of the numerator and denominator could potentially "cancel" them and thus eliminate the asymptote at that point.

• Vertical asymptotes appear as vertical lines on a graph where the function shoots off to infinity.
• The function never touches or crosses a vertical asymptote, manifesting a distinct gap in the graph.

Understanding asymptotic behavior provides insights into the limits and trends of rational functions as they extend towards specific values or infinity. This knowledge is immensely valuable when graphing and analyzing complex functions, supporting deeper analytical studies in mathematics.