If is 1-1 and onto then the correspondence that goes backwards from Y to X is also a function and is called f inverse, denoted . As this is the symbol used for inverse image of a set it must be clear from the context how the symbol is being used. This map is easily described by and if and only if y=f(x). So if f(3)=5 then automatically. If then . In terms of pairs this says if (17,-8) is in the graph of f then (-8,17) is in the graph of .
This relationship is easy to remember for real functions since switching coordinates of a point in the plane puts you at the reflection of the original point about the line y=x. Thus the graph of must be the reflection of the graph of f about the line y=x. This is a great help if the graph of f is already known.
It's the 1-1 condition that is really critical for constructing an inverse function. If f is 1-1 but not onto we can simply replace the codomain with the range f(X) so that is then 1-1 and onto so we can talk about an inverse .
There are some obvious questions that arise here. What are the proper domain and range for an inverse function? Given a formula for f, how does one find a formula for ? Can a formula always be found?
If a function is claimed to be the inverse of a given function, this claim can be checked by a pair of well-known formulas that tell exactly how f and fit together under composition. For 1-1 and onto we always have:
If looks strange at first, keep in mind we used and earlier only to emphasize the fact that we had two elements that came from two possibly different sets. In general, we can call elements of a set by any name we choose. Here we have two separate equations, so it's all right to use in one of them and in the other. It is easy to verify these properties. For the first one, start with letting . We would like to show that z=x. Since is equivalent to f(x)=f(z) by the definition of inverse, we are done as the 1-1 property says then that z=x. The second property has the same proof.