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Chapter 1: 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 help determine if two functions are inverses of each other by checking if the graphs of the two functions are mirror images of each other across the line y=x.

Step by step solution

01

Understanding Inverse Functions

An inverse function is a function that undoes the action of the other function. A function \( f \) and its inverse \(f^{-1}\) are said to be inverses if and only if \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \) where \( f(f^{-1}(x)) \) is the function complement of the inverse function and \( f^{-1}(f(x)) \) is the inverse function complement of the function.
02

Reflection concept

The graphs of a function and its inverse are reflections of each other across the line \( y=x \). If a given point, say, (a, b) lies on the plot of the function, then the point (b, a) will lie on the plot of the inverse function.
03

Plotting the functions

Now, to visually determine if two functions are inverses of each other, first, plot both functions on the same set of axes. Then, also plot the line \(y=x\).
04

Visually checking the reflection

Inspect the resulting plot. If the graphs of the functions are symmetrical about the line \(y=x\), then they are inverses of each other.

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