Problem 44 - Calculus: Early Transcendentals

Question

(a) Find the vertical asymptotes of the function $$ y = \frac{x^2 +1}{3x - 2x^2} $$ (b) Confirm your answer to part (a) by graphing the function.

Step-by-Step Solution

Answer

Vertical asymptotes at \( x = 0 \) and \( x = \frac{3}{2} \).

Step 1: Identify Points of Discontinuity

First, we need to find the points where the function is undefined. This happens when the denominator of the function equals zero. For the function \( y = \frac{x^2 +1}{3x - 2x^2} \), set the denominator equal to zero: \( 3x - 2x^2 = 0 \).

Step 2: Solve the Denominator Equation

Factor the equation \( 3x - 2x^2 = 0 \). This can be rewritten as \( x(3 - 2x) = 0 \). Set each factor to zero to solve for \( x \): - \( x = 0 \)- \( 3 - 2x = 0 \) which gives \( x = \frac{3}{2} \).

Step 3: Determine Vertical Asymptotes

The values of \( x \) that make the denominator zero and do not also make the numerator zero correspond to vertical asymptotes. Check the numerator at \( x = 0 \) and \( x = \frac{3}{2} \). The numerator \( x^2 + 1 \) is never zero for real numbers, so vertical asymptotes occur at both \( x = 0 \) and \( x = \frac{3}{2} \).

Step 4: Confirm Through Graphing

Graph the function \( y = \frac{x^2 + 1}{3x - 2x^2} \). You will see that the graph shows lines approaching infinity at \( x = 0 \) and \( x = \frac{3}{2} \), confirming the vertical asymptotes.

Key Concepts

Points of Discontinuity

In mathematics, particularly in calculus and rational functions, points of discontinuity are crucial to understanding the behavior of functions. Points of discontinuity occur where a function is not defined. For rational functions, like the one given in this exercise, discontinuities arise where the denominator equals zero, as the function can't divide by zero. In our specific example, the denominator is zero at the values that satisfy the equation \( 3x - 2x^2 = 0 \). We factor it to \( x(3 - 2x) = 0 \), identifying the points of discontinuity at \( x = 0 \) and \( x = \frac{3}{2} \). These are precisely where the function will have vertical asymptotes, as the numerator is non-zero at these points.

Rational Functions

Rational functions are expressions created by dividing two polynomials. They are written in the form \( \frac{P(x)}{Q(x)} \) where both \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \) is not zero. The given function \( y = \frac{x^2 +1}{3x - 2x^2} \) is an example of a rational function. The key properties of rational functions include their asymptotic behavior, which is pivotal in their analysis. In these functions, horizontal, vertical, and slant asymptotes describe how the graph behaves as it approaches certain lines. Understanding the behavior near discontinuities helps you predict the appearance and limits of the function. Rational functions can have complex graphs with various asymptotes and intercepts, offering insights into the function's entire structure.

Denominator

The denominator of a rational function, found below the dividing line, holds the key to understanding discontinuities and asymptotic behavior. For the function \( y = \frac{x^2 +1}{3x - 2x^2} \), the denominator \( 3x - 2x^2 \) defines where the function is undefined. To explore discontinuities, set the denominator to zero and solve for \(x\). Here, factoring \( 3x - 2x^2 \) leads to \( x(3 - 2x) = 0 \). Solving this equation, we find \( x = 0 \) and \( x = \frac{3}{2} \). At these points, the function will not have a real output, creating vertical asymptotes. When evaluating rational functions, always consider the denominator's role in shaping the graph. It's generally helpful to perform these steps when analyzing rational functions:

• Identify and fact the denominator.
• Set it equal to zero to find discontinuities.
• Check these potential asymptote/dictoninuity points against the numerator.

Understanding the denominator is essential in mastering rational functions.

Graphing Functions

One of the best ways to understand the behavior of a function is by graphing it. This visual representation helps in confirming theoretical predictions about the function’s behavior, like the presence of vertical asymptotes. In this exercise, graphing the rational function \( y = \frac{x^2 + 1}{3x - 2x^2} \) estimated the locations of these asymptotes at \( x = 0 \) and \( x = \frac{3}{2} \). As the graph approaches these x-values, it becomes clear that there's a vertical asymptote, as the function tends toward infinity.

• Graphing functions can illustrate points of discontinuity and confirm asymptotic behavior.
• It can help verify solutions and present the practical implications of mathematical analysis.

Therefore, learning how to graph functions will solidify your understanding of abstract concepts and provide a holistic view of their behaviors.