# Functions

## Functions A function is a relation between a set of inputs and a set of possible outputs with the property that each input is associated with exactly one output. In other words, a function is a rule that assigns to each input value exactly one output value.

A function is simply used to represent the dependence of one quantity on the other and easily defined with the help of the concept of mapping

A function can be defined as follows:

Let A and B be two non-empty sets. A function f from A to B is a rule that assigns to each element x in A a unique element y in B, denoted by f(x). The set A is called the domain of the function, and the set B is called the codomain or range of the function.

**For example,** the function f(x) = x^{2} maps each real number x in the domain (-∞, ∞) to a unique non-negative real number y in the range [0, ∞).

Functions in mathematics are used to model relationships between variables and to describe mathematical operations. They are used in various branches of mathematics, including calculus, algebra, and geometry. Functions can be analyzed using techniques such as differentiation, integration, and graphing.

A function from set P to set Q is a rule that assigns to each element of set P, one and only one element of set Q.

Mathematically: If f: P->Q where y = f(x), x ∈ P and y ∈ Q. Here y is the image of x under f.