# Functions and Types of Functions: What Are Functions in Mathematics? (For CBSE, ICSE, IAS, NET, NRA 2022)

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# Title: Functions and Types of Functions

- Functions are relations where each input has a particular output.
- We can define a function as a special relation which maps each element of set A with one and only one element of set B. Both the sets A and B must be non-empty.
- In this lesson, the concepts of functions in mathematics and the different types of functions are covered using various examples for better understanding.

## What Are Functions in Mathematics?

- A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
- A function defines a particular output for a particular input. Hence, is a function such that for a ∈ A there is a unique element b ∈ B such that
- Let A & B be any two non-empty sets, mapping from A to B will be a function only when every element in set A has one end only one image in set B.

## Example

- Another definition of functions is that it is a relation f in which each element of set A is mapped with only one element belonging to set B.
- Also, in a function, there cannot be two pairs with the same first element.

## A Condition for a Function

- Set A and Set B should be non-empty.
- In a function, a particular input is given to get a particular output. So, A function denotes that f is a function from A to B, where A is a domain and B is a co-domain.
- Many widely used mathematical formulas are expressions of known functions.
- For an element, a, which belongs to A, a unique element b, is there such that
- For example, the formula for the area of a circle, , gives the dependent variable A (the area) as a function of the independent variable r (the radius) .
- The unique element b to which f relates a, is denoted by f (a) , and is called f of a, or the value of f at a, or the image of a under f.
- The range of f (image of a under f)
- It is the set of all values of f (x) taken together.
- Range of f =
- A real-valued function has either P or any one of its subsets as its range.
- Further, if its domain is also either P or a subset of P, it is called a real function.

## Representation of Functions

- Functions are generally represented as f (x)
- Let,
- It is said as f of x is equal to x cube.