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.
