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Chapter 2: Problem 74

How can a graphing utility be used to visually determine if two functions are inverses of each other?

Short Answer

Expert verified
A graphing utility can be used to visually determine if two functions are inverses of each other by plotting the functions and the line \(y = x\) on the same graph. If the two functions are inverses, their plots will mirror each other over the line \(y = x\).

Step by step solution

01

Understanding the Concept of an Inverse Function

First, there should be a clear understanding of an inverse function. An inverse function is a function that undoes the action of the another function. A function \(g(x)\) is the inverse of a function \(f(x)\) if for every \(x\) in the domain of \(f\), \(g(f(x)) = x\); and for every \(x\) in the domain of \(g\), \(f(g(x)) = x\). It's also important to remember that not all functions have inverse functions.
02

Graphing the Functions

Use the graphing utility to graph both functions. Label the graph of the first function \(f(x)\) and the second function \(g(x)\) for clarity.
03

Checking the Reflection

Draw the line \(y = x\) using the same graphing utility. This line is the mirror line. If the two functions are indeed inverses of each other, their graphs will reflect each other over the line \(y = x\).
04

Visual Confirmation

Visually inspect the graphs. If the graph of \(f\) is a reflection of the graph of \(g\) across the line \(y = x\), then \(f\) and \(g\) are inverse functions of each other. The points on \(f\) that lie above the line \(y = x\) will correspond to points on \(g\) that lie below it and vice versa.

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