Functions are relations where each input has a particular output. In this lesson, the concepts of functions in mathematics and the different types of functions are covered using various examples for better understanding.

**Contents Related to Functions**

## 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. 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 can’t 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 **f: A->B** denotes that f is a function from** A** to** B**, where **A** is a domain and **B** is a co-domain.

- For an element,
**a**, which belongs to**A**,$a\epsilon A$ , a unique element**b**,$b\epsilon B$ is there such that (a,b)$\epsilon$ f.

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**= { y$\epsilon$ Y | y = f (x), for some x in X}

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**.

**Vertical Line Test:**

Vertical line test is used to determine whether a curve is a function or not. If any curve cuts a vertical line at more than one points then the curve is not a function.

### Representation of Functions

Functions are generally represented as

Let ,

It is said as f of x is equal to x cube.

Functions can also be represented by g(), t(),… etc.

### Steps for Solving Functions

**Question:** Find the output of the function

(i) t = 0

(ii) t = 2

**Solution:**

The given function is

(i) At t = 0,

= 5

(ii) At t = 2,

= 29

### Functions and Relations Video

## Types of Functions

There are various types of functions in mathematics which are explained below in detail. The different functions types covered here are:

- One – one function (Injective function)
- Many – one function
- Onto – function (Surjective Function)
- Into – function
- Polynomial function
- Linear Function
- Identical Function
- Quadratic Function
- Rational Function
- Algebraic Functions
- Cubic Function
- Modulus Function
- Signum Function
- Greatest Integer Function
- Fractional Part Function
- Even and Odd Function
- Periodic Function
- Composite Function
- Constant Function
- Identity Function

### One – one function (Injective function)

If each element in the domain of a function has a distinct image in the co-domain, the function is said to be one – one function.

**For examples** f; R R given by f(x) = 3x + 5 is one – one.

### Many – one function

On the other hand, if there are at least two elements in the domain whose images are same, the function is known as

many to one.

**For example** f : R R given by f(x) = x^{2} + 1 is many one.

### Onto – function (Surjective Function)

A function is called an onto function if each element in the co-domain has at least one pre – image in the domain.

### Into – function

If there exists at least one element in the co-domain which is not an image of any element in the domain then the

function will be Into function.

(Q) Let A = {x : 1 < x < 1} = B be a mapping f : A B, find the nature of the given function (P) F(x) = |x|

f (x) = |1|

Solution for x = 1 & -1

Hence it is many one the Range of f(x) from [-1, 1] is

[0,1] which is not equal to co-domain. Hence it is into function.(R) f (x) = x(x)

**Solution:**

For different values of Input, we have different output hence it is one – one function also it manage is equal to its co-domain hence it is onto also.

### Polynomial function

A real valued function** f : P → P** defined by** y = f (a)=****n** ** N**, and **P**, for each **a** \ **P**, is called p**olynomial function. **

**N**= a non-negative integer.- The
**degree**of Polynomial function is the highest power in the expression. - If the
**degree is****zero**, it’s called a constant function. - If the
**degree is on**e, it’s called a linear function. Example: b = a+1. - Graph type: Always a straight line.

So, a polynomial function can be expressed as :

The highest power in the expression is known as the **degree of the polynomial function**. The different types of polynomial functions based on the degree are:

- The polynomial function is called as Constant function if the degree is zero.
- The polynomial function is called as Linear if the degree is one.
- The polynomial function is Quadratic if the degree is two.
- The polynomial function is Cubic if the degree is three.

### Linear Function

All functions in the form of ax + b where a,

For example, f(x) = 2x + 1 at x = 1

f(1) = 2.1 + 1 = 3

f(1) = 3

Another example of linear function is y = x + 3

### Identical Function

Two** functions f and g are said to be identical** if

**(a)** The domain of f = domain of g

**(b)** The range of f = the Range of g

**(c)**

**For example** f(x) = x & g(x) =

Solution: f(x) = x is defined for all x

But g(x) =

Hence it is identical for

### Quadratic Function

All function in the form of y = ax^{2} + bx + c where a, b,

At

In simpler terms,

A Quadratic polynomial function is a second degree polynomial and it can be expressed as;

F(x) = ax^{2} + bx + c, and a is not equal to zero.

Where a, b, c are constant and x is a variable.

Example, f(x) = 2x^{2} + x – 1 at x = 2

If x = 2, f(2) = 2.2^{2} + 2 – 1 = 9

**For Example**: y = x^{2} + 1

**Read More: **Quadratic Function Formula

### Rational Function

These are the real functions of the type **f (a)** and** g (a)** are polynomial functions of **a** defined in a domain, where **g(a) ≠ 0.**

- For example
**f : P – {– 6} → P**defined by**f (a) =**$\frac{f(a+1)}{g(a+2)}$ ,$\forall a\epsilon$ P – {–6 }is a rational function. - Graph type: Asymptotes (the curves touching the axes lines).

### Algebraic Functions

A function that consists of a finite number of terms involving powers and roots of independent variable x and fundamental operations such as addition, subtraction, multiplication, and division is known as an algebraic equation.

For Example,

### Cubic Function

A cubic polynomial function is a polynomial of degree three and can be expressed as;

F(x) = ax^{3} + bx^{2} + cx + d and a is not equal to zero.

In other words, any function in the form of f(x) = ax^{3} + bx^{2} + cx + d, where a, b, c,

**For example:** y = x^{3}

Domain

Range

### Modulus Function

The real function **f : P → P** defined by f (a) =

**0, and f(a) = -a if a<0**

*modulus function.*

- Domain
*of**f**=**P* - Range of
**f**=$P^{+}\cup {0}$

Domain: R

Range: [0,

### Signum Function

The real function **f : P → P** defined by

is called the** signum function** or **sign function**.(gives the sign of real number)

- Domain of f =
**P**, - Range of f = {1, 0, – 1}

For example: signum (100) = 1

signum (log 1) = 0

signum (x^{2}1) =1

### Greatest Integer Function

The real function **f : P → P** defined by** f (a) = [a], a** \ **P** assumes the value of the greatest integer less than or equal to **a,** is called the** greatest integer function. **

- Thus f (a) = [a] = – 1 for – 1 ⩽ a < 0
- f (a) = [a] = 0 for 0 ⩽ a < 1
- [a] = 1 for 1 ⩽ a < 2
- [a] = 2 for 2 ⩽ a < 3 and so on…

The greatest integer function always gives integral output. The Greatest integral value that has been taken by the input will be the output.

For example: [4.5] = 4

[6.99] = 6 [1.2] = 2Domain

Range

### Fractional Part Function

{x} = x – [x]

It always give fractional value as output.

**For example**:- {4.5} = 4.5 – [4.5]

= 4.5 – 4 = 0.5

{6.99} = 6.99 – [6.99]

= 6.99 – 6 = 0.99

{7} = 7 – [7] = 7 –7 = 0

### Even and Odd Function

If f(x) = f(-x) then the function will be even function & f(x) = -f(-x) then the function will be odd function

**Example 1:**

f(x) = x^{2}sinx f(x) = -f(-x)

f(-x) = -x^{2}sinx

it is odd function.

**Example 2:**

### Periodic Function

A function is said to be a periodic function if there exist a positive real numbers T such that f(u – t) = f(x) for all x ε Domain.

For example f(x) = sinx

f(x + 2π) = sin (x + 2π) = sinx fundamental

then period of sinx is 2π

### Composite Function

Let A, B, C’ be three non-empty sets

Let f: A B & g : G C be two functions then gof : A C. This function is called composition of f and g

given g of (x) = g(f(x))

For example f(x) = x^{2} & g(x) = 2x

f(g(x)) = f(2x) = (2x)^{2} = 4x^{2}

g(f(x)) = g(x^{2}) = 2x^{2}

### Constant Function

The function **f : P → P** defined by **b = f (x) = D**, **a** **P**, where **D** is a constant **P**, is a constant function.

- Domain of
**f = P** - Range of
**f = {D}** - Graph type: A straight line which is parallel to the x-axis.

In simple words, the polynomial of 0th degree where f(x) = f(0) =

### Identity Function

**P**= set of real numbers

The function **f : P → P** defined by **b = f (a) = a** for each a **P** is called the identity function.

- Domain of
**f**=**P** - Range of
**f**=**P** - Graph type: A straight line passing through the origin.