Graphing an Inverse Map

Given a function  f : A → B we define the inverse map  f ^-1: B → A by saying that  f ^-1(b) = a means  f (a) = b.

In terms of the graph, this tells us that the point (b,a) is on the graph of the map  f ^-1  whenever the point (a,b) is on the graph of  f.

To make a graph of  f ^-1 by hand:

• Start with a graph of  y = f (x) and a second, blank x,y axes for the graph of f ^-1.
• Find a key point (x,y) = (a,b) on the graph of  f, and interchange the x,y coordinates to get a new ordered pair (b,a).
• Plot the ordered pair (b,a) on the axes for  f ^-1.
• Repeat this for a bunch of key points.
• When you have enough key points plotted, sketch the graph of  f ^-1.
• (The steps above are equivalent to doing the following: Start with a graph of y = f (x) and flip the graph across the line y = x to obtain the graph of  y = f ^-1(x).)

To make a graph of  f ^-1 by computer:

• Find a good graphing utility, such as Desmos.com.
• Suppose that you have a formula for  f . That is  f (x) = some expression involving x.
• Type the equation y = some expression involving x into the graphing utility to get a graph of  f.
• Interchange the x,y in your equation to get a new equation of the form x = some expression involving y. Type this new equation into the graphing utility to get a graph of  f ^-1.
• (If you type both equations into Desmos, you will get both a graph of  f  and a graph of  y = f ^-1.)