 A function (or map) is a rule or correspondence that associates each element of a set X called the domain with a unique element of another set Y called the codomain. We typically give the rule a name such as a letter like f or g (or any letter of your choice) or a name agreed upon by convention like sine or log or square root. The term ''unique'' is critical in the definition as this says that one  cannot be associated with 2 or more elements of Y. Thus we can call the y value corresponding to a particular x under the rule f by the name f(x) (read f of x) since there is only one such y. However, associating 2 elements of X with the same  and some elements of Y not corresponding to any element of X are allowed as they are not ruled out in the definition. The fact that the correspondence goes from the domain to the codomain is indicated by the notation  . 
For a subset  we use f(A) to mean the set  called the image of A. (Some authors use  .) The set f(X) is called the range of f. While it is always the case that  , in general they need not be equal. It is important to distinguish while  . For a subset  we use  to mean the set  called the inverse image of B. (Some authors use  .) Keep in mind that while  .  If and  we can form a new function called a composite map denoted by  . The symbol  is read as ''g circle f.'' The composition is accomplished by defining  for each . 
 A function is called an injection or one-to-one (also written 1-1) if different x values get mapped to different y values. So if  then  . this is equivalent to the contrapositive, if  then  . We call f a surjection or onto if every corresponds to some , that is, f(X)=Y. A bijection is used for functions that are both 1-1 and onto. The set of ordered pairs  is the graph of f. There are many ways to describe or write functions. The method used depends on many factors such as the domain of the function, the complexity of the function or even the subject matter in which the function arises. We describe a few common methods here. (1) If the domain is small we might just list all function values. Let  and  . If I tell you that f(1)=b, f(2)=b and f(3)=d then I have completely defined a function . Observe that this f neither 1-1 nor onto and has range  , a proper subset of Y. (2) We could list all pairs in the graph of f, again if the domain is small. The function in (1) could be given by  .  (3) Quite often simple functions like the example in (1) are indicated by just listing the elements of both the domain and codomain and connecting x values to function values by arrows. Here's how that's done.
(4) For numerical functions, a formula might be given in terms of x that calculates the value of f(x). For X=Y=R the set of real numbers,  would define  . Since  , the point (2,7) would be a point in the graph but there certainly are too many points to list as in (2).  (5) A picture of the graph in (4) would be a good way to visually represent f instead of just supplying the formula. The picture is simple a plot of the points (x,y) in the plane  that satisfy the equation  .
(6) It is sometimes possible to describe a function in words without resorting to lists, formulas or pictures. For example, the air temperature where you are sitting right now can be considered a function of time. That is, to each point in time we associate the corresponding temperature at that time. The domain consists of whatever scale is used to record time (12 or 24 hr. clock, minutes or fractions of hours, etc.) and the range is a subset of the real numbers (whatever temperature scale is used). This is a function by definition, yet you probably don't have a formula for this correspondence (well, unless the temperature is constant where you are).  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. | 