In most calculus books a function is introduced as a rule that assigns to each object in a certain set (domain) a unique object in a possibly different set (range).
Typical example:
The action of this function can be described by writing,
Each individual action can be represented by an ordered pair:
The set of all these pairs represents the squaring function. These set of pairs at times has been called the graph of the function; it is a subset of the coordinate plane 
Thus a function is a set of ordered pairs (i.e., a relation). But it has a special property: it is "single-valued" such that for each domain there is a unique y such that
Definition A function is a relation F such that for each x in dom F there is only one y such that
For a function F and a point x in dom F, the unique y such that 
The "
Functions are basic objects appearing in all parts of mathematics. Thus, there are many terminologies associated with functions. Unfortunately, no terminology has become uniformly standardized. In this section, we will collect some of this terminology.
Mapping
F is a function from A into B or that F maps A into B written as,
iff F is a function, dom F = A, and ran 
A special case of range such that ran F = B, then F is a function from A onto B.
Injection
A function F is one-to-one iff for each 
Definition A set R is single-rooted iff for each 
We can then say that a function is single-rooted iff it is one-to-one.
Consider the following function called "addition", 
We may write the function as
Now, we'll discuss some commonly applied operations to functions, using some defined arbitrary sets
Definition
a. The inverse of F is the set
b. The composition of F and G is the set
c. The restriction of F to A is the set
d. The image of A under F is the set
Facts about Functions
Theorem 3E For a set F, dom 
Theorem 3F For a set F, 
Theorem 3G Assume that F is a one-to-one function. If 
Theorem 3H Assume that F and G are functions. Then 
and for x in its domain, 
The thing about our definition is that we want them to be general such that they are applicable even to nonfunctions.
Theorem 3I For any sets F and G,
This theorem expresses common knowledge. In getting dressed, one first put on socks and then shoes. But in the inverse process of getting undressed, one first removes shoes and then socks.
Theorem 3J Assume that 
a) There exists a function 
b) There exists a function 
Axiom of Choice: (first form) For any relation there is a function 
Corollary 3L For any function 
We conclude our discussion of functions with some definitions that may be useful later. Out intent is to build a large working vocabulary of set-theoretic notations.
Consider an infinite union 
Example: if 
Disclaimer: this is a summary of section 3.4 from the book "Elements of Set Theory" by Herbert B. Enderton, the content apart from rephrasing is identical, most of the equations are from the book and the same examples are treated. All of the equation images were screenshots from generated latex form using typora
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