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Chapter 2: Problem 73

Describe how to use the graph of a one-to-one function to draw the graph of its inverse function.

Short Answer

Expert verified
The graph of a function and its inverse can be obtained by reflecting the function's graph over the line y=x, because the x-coordinates and y-coordinates are interchanged for a function and its inverse.

Step by step solution

01

Understand the properties of inverse functions

Understand that for a function \(f(x)\), and its inverse \(f^{-1}(x)\), for all x in the domain of \(f(x)\) and \(f^{-1}(x)\), we have \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). This means the output of a function at a particular x-coordinate will become the input of the inverse function.
02

Draw the line y=x on the graph

Identify the line y=x on the Cartesian plane. This line will serve as the mirror upon which to reflect the graph of the function to plot the graph of the inverse function.
03

Reflect the graph across the line y=x

Reflect the graph of the given one-to-one function across the line y=x. The reflected graph is the graph of the inverse function.
04

Verify the graph of the inverse function

Check whether the graph of the inverse function has been accurately drawn. A correct graph will intersect the line y=x exactly where the original function had a horizontal tangent, and will lie entirely within the first and third quadrants if the original function does.

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