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.)