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 itÂ is a relation “f” in which each element of setÂ “A” is mapped with only one element belonging to set “B”. Also in a function, there canâ€™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**fÂ**(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 Â 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 \(f(x)\)

Let , \(f(x)= x^{3}\)

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 \(g(t)= 6t^{2}+5\) at

(i) t = 0

(ii) t = 2

**Solution:**

The given function is \(g(t)= 6t^{2}+5\)

(i) At t = 0, \(g(0)= 6(0)^{2}+5 \)

= 5

(ii) At t = 2, \(g(2)= 6(2)^{2}+5 \)

= 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 â€“ one function (Injective function)
- Many â€“ one function
- Onto â€“ function (Surjective Function)
- Into â€“ 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 â€“ one function.

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

### Many â€“ 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 â€“ 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 â€“ 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:** \(f(x)=\left\{\begin{matrix} x^2 & ; & x\geq 0\\ -x^2 & ; & x<0 \end{matrix}\right.\)

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

### Polynomial function

A real valued function** f : P â†’ P** defined by** y = f (a)=**\(h_{0}+h_{1}a+…..+h_{n}a^{n}\), where **n** \(\epsilon\)** N**, and \(h_{0}+h_{1}+…..+h_{n}\)\(\epsilon\) **P**, for each **a** \ \(\epsilon\) **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**, itâ€™s called a constant function. - If the
**degree is on**e, itâ€™s called a linear function. Example: b = a+1. - Graph type: Always a straight line.

So, a polynomial function can be expressed as :

\(f(x)= a_{n}x^{n}+a_{n-1}x^{n-1}+…..+a_{1}x^{1}+a_{0}\)

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, \(b\in R\) & a â‰ 0 are called as linear functions. The graph will be a straight line. In other words, aÂ linear polynomial function is a first-degree polynomial where the input needs to be multiplied by m and added to c. It can be expressed by f(x) = mx + c.

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

f(1) = 2.1 + 1 = 3

f(1) = 3

Another example of linear function isÂ 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)** \(f\left( x \right)\text{ }=\text{ }g\left( x \right)\forall x\in {{D}_{f}}\And \text{ }{{D}_{g}}\)

**For example** f(x) = x & g(x) =\(\frac{1}{{}^{1}/{}_{x}}\)

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

But g(x) = \(\frac{1}{{}^{1}/{}_{x}}\) is not defined of x = 0

Hence it is identical for \(x(~\in R\text{ }\text{ }\left\{ 0 \right\}\)

### Quadratic Function

All function in the form of y = ax^{2} + bx + c where a, b, \(c\in R\), a â‰ 0 will be known as Quadratic function. The graph will parabolic.

At \(x=\frac{-b \pm \sqrt{D}}{2}\), we will get its maximum on minimum value depends on the leading coefficient and that value will be \(\frac{-D}{4a}\) (where D = Discriminant)

In simpler terms,

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

F(x) = ax^{2}Â + 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}Â + x â€“ 1 at x = 2

If x = 2, f(2) = 2.2^{2}Â + 2 â€“ 1 = 9

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

**Read More: **Quadratic Function Formula

### Rational Function

These are the real functions of the type \(\frac{f(a)}{g(a)}\) where **f (a)** and** g (a)** are polynomial functions of **a** defined in a domain, where **g(a) â‰ 0.**

- For example
**f : P â€“ {â€“ 6} â†’ P**defined by**f (a) =**\(\frac{f(a+1)}{g(a+2)}\), \(\forall a\epsilon\)P â€“ {â€“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,

\(f(x)=5x^{3}-2x^{2}+3x+6\), \(g(x)=\frac{\sqrt{3x+4}}{(x-1)^{2}}\).

### Cubic Function

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

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

In other words, any function in the form of f(x) = ax^{3}Â + bx^{2}Â + cx + d, where a, b, c, \(d\in R\) & a â‰ 0

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

Domain\(\in\)R

Range\(\in\)R

### Modulus Function

The real function **f : P â†’ P** defined by f (a) = \(f(a)=\left | a \right |\) = a, a\(\geq\)

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

*modulus function.*

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

Domain: R

Range: [0, \(\infty\))

### Signum Function

The real function **f : P â†’ 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, â€“ 1}

For example: signum (100) = 1

signum (log 1) = 0

signum (x^{2}1) =1

### Greatest Integer Function

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

- Thus f (a) = [a] = â€“ 1 for â€“ 1 â©½ a < 0
- f (a) = [a] = 0 for 0 â©½ a < 1
- [a] = 1 for 1 â©½ a < 2
- [a] = 2 for 2 â©½ a < 3 and so onâ€¦

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\(\in\)R

Range\(\in\)Integers

### Fractional Part Function

{x} = x â€“ [x]

It always give fractional value as output.

**For example**:- {4.5} = 4.5 â€“ [4.5]

= 4.5 â€“ 4 = 0.5

{6.99} = 6.99 â€“ [6.99]

= 6.99 â€“ 6 = 0.99

{7} = 7 â€“ [7] = 7 â€“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 Îµ Domain.

For example f(x) = sinx

f(x + 2Ï€) = sin (x + 2Ï€) = sinx fundamental

then period of sinx is 2Ï€

### Composite Function

Let A, B, Câ€™ 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 â†’ P** defined by **b = f (x) = D**, **a** \(\epsilon\)**P**, where **D** is a constant \(\epsilon\) **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) = \(a_{0}\)=c. Regardless of the input, the output always results in constant value. The graph for this is a horizontal line.

### Identity Function

**P**= set of real numbers

The function **f : P â†’ P** defined by **b = f (a) = a** for each a \(\epsilon\) **P** is called the identity function.

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