Definition of Function

Definition of function

Standard textbook definition

Let A and B be nonempty sets. A function f from A to B is a rule that assigns to each element a of set A one and only one element, called f(a), from set B.

Alternate definition

A function must satisfy three requirements

(1) Start with a pair of not necessarily distinct nonempty sets usually designated as the first set and the second set,

(2) Each element of the first set is assigned a partner from the second set, and

(3) No element from the first set is assigned two or more partners from the second set.

Definition in terms of ordered pairs

Given sets A and B, the Cartesian product of A with B, denoted A x B, is defined as A x B = {(a,b) | a is in A and b is in B}.

Given sets A and B, a relation from A to B is a subset of the Cartesian product of A with B.

A function f from A to B is a relation from A to B for which:

(1) A and B are not necessarily distinct nonempty sets,

(2) for each element a in A, there is an element b of B such that (a,b) is in f,

(3) for all a in A and for all b and c in B,

if (a,b) is in f and (a,c) is in f, then b = c

(2.alternate wording)

Each element of A is a first coordinate of some ordered pair of f.

(3.alternate wording)

No element of A is a first coordinate of more than one ordered pair in f.

Radian Measure

Place the vertex of an angle at the center of a circle with radius r. Let s be the arc length of the circle subtended by the angle. The radian measure q of the angle is given by provided that r and s are measured in the same linear units.

Circular Trigonometric Functions

Let q be a measure of an angle in standard position. Let P with coordinates (x,y) be the point of intersection of the terminal side of the angle with the unit circle x2 + y2 = 1. Then, define the (circular) trigonometric functions of q by