Problem 56 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$f(x)=-\frac{\left|x^{2}-9\right|}{\left|x^{2}+7\right|}$$

Step-by-Step Solution

Answer

After graphing the function and conducting a horizontal line test, it can be determined whether the function \(f(x)=-\frac{\left|x^{2}-9\right|}{\left|x^{2}+7\right|}\) is one-to-one, and if it has an inverse function.

Step 1: Graph the function

First, graph the given function \(f(x)=-\frac{\left|x^{2}-9\right|}{\left|x^{2}+7\right|}\) using a graphing program or utility.

Step 2: Conduct the Horizontal Line Test

Draw horizontal lines through the graph in various places. If any of these horizontal lines intersects the graph in more than one point, then the function is not one-to-one and thus wouldn't have an inverse function.

Step 3: Analysis of the Graph

Analyse the results from the horizontal line test. If it passes the horizontal line test (each line intersects the graph at most once), then the function is said to be one-to-one, which means it has an inverse function. If the graph fails to meet the condition of the horizontal line test, then the function is not one-to-one, hence, does not have an inverse.

Key Concepts

Horizontal Line Test

Understanding the Horizontal Line Test is crucial when studying functions and their inverses. Simply put, the Horizontal Line Test is a visual method of determining if a function is one-to-one; that is, whether each output from the function is the result of only one input.

If you can draw a horizontal line anywhere across a graph of the function and it crosses the graph more than once, then the function does not pass the test and is not one-to-one. Failing this test means that the function does not have an inverse, since for inverse functions, each output must be tied to a single, unique input.

For the exercise in question, applying the Horizontal Line Test to the graph of the function \(f(x)=-\frac{\left|x^{2}-9\right|}{\left|x^{2}+7\right|}\) would inform us whether an inverse function can exist for this particular set of rules.

Inverse Functions

Inverse functions are pairs of functions that 'undo' each other. If you have a function that takes you from 'A to B', an inverse function takes you back from 'B to A'. The notation usually changes from \(f(x)\) to \(f^{-1}(x)\) to represent the inverse.

For a function to have an inverse, it must pass the Horizontal Line Test, meaning it must be one-to-one. In terms of graphing, if \(f(x)\) has a point \(x, y\), then its inverse \(f^{-1}(x)\) has a point \(y, x\). The graph of an inverse function is a reflection across the line \(y=x\).

It's important to note that some functions, such as quadratics and absolute value functions, are not one-to-one over their entire domain, but they can be restricted to a portion of their domain where they are one-to-one, allowing for an inverse on that restricted domain.

Graphing Utility

A graphing utility is an essential tool for any student in mathematics. It allows you to input a function and visualize its graph, providing immediate feedback about its properties. Especially when dealing with complex functions or those involving absolute values, graphing utilities reveal important aspects like symmetry, intercepts, and potential barriers.

For the given exercise, the graphing utility would help you not just to draw the graph but also to apply the Horizontal Line Test, remove any guesswork, and ensure accuracy in your conclusion about whether the function has an inverse.

Absolute Value

The absolute value of a number is the distance of the number from zero on the number line, irrespective of direction. The notation used is \( |x| \), where \(x\) is any real number. Grasping this concept is fundamental when dealing with absolute value functions, like the one in our exercise, \(f(x)=-\frac{\left|x^{2}-9\right|}{\left|x^{2}+7\right|}\).

One intricacy of absolute value functions is that they can introduce non-one-to-one behaviour, because they result in the same value for positive and negative inputs. For example, \( |2| = |-2| = 2 \). This can complicate the process of finding an inverse, because for some ranges of \(x\), the function might fail the Horizontal Line Test.