Inverse functions
A function turns inputs into outputs. When can it be run backwards, and how do you find the function that does it?
Where you are 5 inside
What this area is about
An inverse function undoes another: if f sends a to b, its inverse f^{-1} sends b back to a. Not every function has one. It works only when no two inputs share an output, and this area covers that test, how to find an inverse from a table, a formula or a graph, and what to do when a function fails the test.
How these fit together
When an inverse exists. One to one functions and the horizontal line test is the condition: each output comes from a single input.
Procedures by form. Inverse from a table swaps the input and output columns. Finding an inverse from a rule swaps the roles of x and y and solves.
A result. The graph of an inverse is the original graph reflected across the line y = x.
A repair. Restricting a domain keeps a one to one piece of a function, such as half a parabola, so that piece can be inverted.
A path through it
One to one functions first, then inverses from a table, then from a rule. After the rule come the graph of an inverse and restricting a domain, in either order.
Next, the inverse of the exponential is the logarithm, drawn in graphs of logarithmic functions, and restricting the domain of a repeating function makes the inverse trigonometric functions possible.