Given two sets 
and 
, we call a function the correspondence from to in which all elements of have at most one image in , that is, one image or none.
A real function of a real variable is any correspondence 
that associates to each element of a certain subset of real numbers, called the domain, another real number.
The subset in which the function is defined is called the domain or field of existence of the function. It is designated by 
.
The number 
belonging to the domain of the function is called the independent variable.
The number 
, associated by to the value , is called the dependent variable. The image of is designated by 
.
Therefore:
The range (or codomain) of a function is the set of real values that the variable or takes.
Initial set Final set
Domain Image set or range
The domain is the set of elements that have an image.
The range is the set of elements that are images.
Composition of Functions
If we have two functions: and 
, such that the domain of the second is included in the range of the first, we can define a new function that associates to each element of the domain of the value of 
.
Domain
Properties
1 Associative.
2 Not commutative.
3 Identity element is the identity function, 
.
Examples of Function Composition
Consider the functions:
1
2
3
Inverse or Reciprocal Function
The inverse or reciprocal function of is another function 
that satisfies:
If 
, then 
.
We can observe that:
- If two functions are inverse, their composition is the identity function.
- The domain of is the range of .
- The range of is the domain of .
- If we want to find the range of a function, we must find the domain of its inverse function.
The graphs of and are symmetric with respect to the bisector of the first and third quadrants.
It is important to distinguish between the inverse function, 
, and the reciprocal of a function, 
.
Steps for Calculating the Inverse Function
1 Write the equation of the function in terms of and .
2 Solve for the variable in terms of the variable .
3 Interchange the variables.
Examples of Calculating the Inverse Function
1
First, we write the equation of the function in terms of and .
We perform the operations.
We verify the result for 
:
2
3
This is not a function.
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