Problem 45 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function. What do you observe about its asymptotes? $$g(x)=\frac{4|x-2|}{x+1}$$

Step-by-Step Solution

Answer

The function has a jump discontinuity at \(x=2\), and has a vertical asymptote at \(x=-1\) and a horizontal asymptote at \(y=0\).

Step 1: Split the Absolute Value Function

Recognize that the absolute function \(|x-2|\) can be split into two functions \(x-2\) and \(-(x-2)\) based on its property. This gives us two separate functions: \(g_1(x)=\frac{4(x-2)}{x+1}\) for \(x\geq2\) and \(g_2(x)=\frac{-4(x-2)}{x+1}\) for \(x<2\).

Step 2: Determine the Asymptotes for Each Function

Identify the vertical asymptote by setting the denominator equal to zero and solving for x, this gives \(x=-1\). The horizontal asymptotes can be determined by looking at the degrees of the numerator and the denominator. Since the degree of the numerator is lower, the horizontal asymptote is \(y=0\). This applies to both \(g_1(x)\) and \(g_2(x)\).

Step 3: Graph the Functions

Gather the information from the previous steps, and draw the graphs of both the functions \(g_1(x) \text{ and } g_2(x)\) for their respective domains. Make sure to clearly mark the asymptotes.

Key Concepts

Asymptotes

When graphing functions, especially rational ones, we encounter unique lines called asymptotes. These are lines that the graph approaches but never actually touches or crosses. There are two main types of asymptotes: vertical and horizontal.

A vertical asymptote occurs when the function approaches infinity or negative infinity as the input approaches a certain value. For the function \(g(x)=\frac{4|x-2|}{x+1}\), we find a vertical asymptote at \(x=-1\) because the denominator becomes zero and the function is undefined. To graph this, draw a dashed line along \(x=-1\) which the graph will approach but not cross.

The horizontal asymptote is a bit different. It describes the behavior of a function as the input grows very large or very small. For the same function, \(g(x)\), because the degree of the polynomial in the numerator is less than that in the denominator, the horizontal asymptote is at \(y=0\). Hence, as \(x\) goes to infinity or negative infinity, the value of \(g(x)\) approaches zero.

Piecewise Functions

A piecewise function is a function that is defined by multiple sub-functions, each applicable to a certain interval of the main function's domain. In the case of absolute value functions, like \(g(x)=\frac{4|x-2|}{x+1}\), it's necessary to consider the function piecewise because the absolute value expression behaves differently depending on the input.

To graph \(g(x)\), you need to split it into two separate linear functions at \(x=2\). If \(x\) is greater than or equal to 2, the function is \(g_1(x)=\frac{4(x-2)}{x+1}\). If \(x\) is less than 2, the function is \(g_2(x)=\frac{-4(x-2)}{x+1}\). This division creates a graph that is a mirror image across the line \(x=2\), illustrating the 'V' shape that is characteristic of absolute value functions. It is essential to ensure that each piece of the function matches up at \(x=2\).

Rational Functions

Rational functions are functions that can be expressed as the ratio of two polynomials, such as \(g(x)=\frac{4|x-2|}{x+1}\). The important properties to note when graphing a rational function include the location of its asymptotes, intercepts, and its overall shape or behavior.

For instance, when the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator, the x-axis (\((y=0\)) is a horizontal asymptote. This relationship tells us about the long-term behavior of the function. Additionally, where the denominator equals zero indicates potential vertical asymptotes, which inform us about the function's undefined regions.

Approaching graphing with these guidelines helps build the framework of the rational function's graph. Understanding the nature of the polynomials within the numerator and denominator is imperative to depicting the correct graphical representation of a rational function.

Graphing Calculators

Using a graphing calculator is a powerful way to visualize functions, especially when dealing with complex functions like absolute value and rational expressions. Graphing calculators allow you to input equations and display their graphs accurately.

For a function like \(g(x)=\frac{4|x-2|}{x+1}\), a graphing calculator can help confirm your hand-drawn sketch or provide a visual when the algebraic manipulation seems challenging. It can also reveal the behavior near the asymptotes and whether the function is continuous or has points of discontinuity.

To use a graphing calculator effectively, you will need to understand how to input piecewise functions, how to denote absolute values, and how to set appropriate viewing windows to capture key features like intercepts and asymptotes. They're an invaluable tool not just for verifying your work but also for learning and intuition-building about how functions behave graphically.