Problem 19 - Calculus of a Single Variable

Question

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ f(x)=\frac{3}{x^{2}+x-2} $$

Step-by-Step Solution

Answer

The vertical asymptotes for the function \(f(x)=\frac{3}{x^{2}+x-2}\) are \(x=-2\) and \(x=1\)

Step 1: Set denominator equal to zero

To find the vertical asymptotes of the function \(f(x)=\frac{3}{x^{2}+x-2}\), set the denominator equal to zero and solve for \(x\). Hence we solve the equation \(x^{2}+x-2=0\)

Step 2: Factor the equation

The next step is to factor the quadratic equation. The factored form of the equation \(x^{2}+x-2=0\) is \((x-1)(x+2)=0\)

Step 3: Find the roots of equation

Now, set each factor equal to zero and solve for \(x\). So, \(x-1=0\) gives \(x=1\) and \(x+2=0\) gives \(x=-2\). These two values of \(x\) make the denominator of the function equal to zero.

Key Concepts

Rational Functions

A rational function is any function that can be expressed as a fraction with two polynomials: one in the numerator and another in the denominator. In the function given, \(f(x) = \frac{3}{x^{2} + x - 2}\), 3 is the constant numerator, and \(x^{2} + x - 2\) is the polynomial denominator. Rational functions have special points known as vertical asymptotes where the denominator equals zero, causing the function to be undefined. These asymptotes reflect the function's behavior as it approaches these values, essentially describing parts of the graph that extend towards infinity. Understanding how to find vertical asymptotes is crucial because they help in graphing rational functions and predicting trends across variable values.

Polynomial Factoring

Polynomial factoring is a key process in algebra and calculus where a polynomial is rewritten as a product of simpler polynomials. To find vertical asymptotes, you need to factor the polynomial in the denominator of a rational function. In our example, we start with the polynomial \(x^{2} + x - 2\). Factoring involves finding two numbers that multiply to \(-2\) and add to \(1\).

These two numbers are \(-1\) and \(2\), leading to the factors \((x - 1)\) and \((x + 2)\). Thus, \(x^{2} + x - 2\) can be factored to \((x - 1)(x + 2)\). This process moves us closer to identifying the critical points that can cause our rational function to approach vertical asymptotes.

Quadratic Equations

A quadratic equation is one of the simplest forms of polynomial equations with the general form \(ax^{2} + bx + c = 0\). When solving rational functions for vertical asymptotes, modifications of quadratic equations often appear in the denominator.

In the function \(f(x) = \frac{3}{x^{2} + x - 2}\), the denominator represents a quadratic equation. Solving it involves factoring, though other methods include using the quadratic formula, completing the square, or graphing. The aim is always to find \(x\) values that make the denominator zero, which correspond to potential vertical asymptotes.

Asymptotic Analysis

Asymptotic analysis involves examining the behavior of functions as the input approaches a particular value or infinity. For rational functions, vertical asymptotes are a crucial part of this analysis.

Vertical asymptotes indicate values that a function nears but never actually reaches as \(x\) approaches certain points from either side. In the problem context, once the denominator \(x^{2} + x - 2\) is factored to \((x - 1)(x + 2)\), setting each factor to zero provides the asymptotic x-values: \(x = 1\) and \(x = -2\).

These vertical asymptotes are significant in understanding the graph's end-behavior, ensuring a comprehensive picture of the function's overall dynamics.