A Review of Functions

General Definition: A real-valued function f defined on a set DA of real numbers is a rule that assigns to each x in D exactly one real number, denoted f(x).

Mapping Definition: A function is a relation that is one to one or many to one but never one to many.

Set Definition: A function is a set of ordered pairs such that no two or more ordered pairs have the same first element.

Domain: The set D of elements for which f(x) is defined.

Range: The set R of elements for which D maps to.

Domain and Range for:

A function maps from set A into set B. If R is all of B then the function is onto. A function that is one to one and onto is said to be a one to one correspondence.

A function must be a one to one in order to have an inverse function!

Example of a one to one correspondence and its inverse:

Example of a function that is not a one to one correspondence for the set of real numbers:

Example of a relation that is not a function: