Problem 73 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.