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# Functions and Types of Functions

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.

## 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,
$$\begin{array}{l}a\epsilon A\end{array}$$
, a unique element b,
$$\begin{array}{l}b\epsilon B\end{array}$$
is there such that (a,b)
$$\begin{array}{l}\epsilon\end{array}$$
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 (image of a under f)
• It is the set of all values of f(x) taken together.
• Range of f = { y
$$\begin{array}{l}\epsilon\end{array}$$
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

$$\begin{array}{l}f(x)\end{array}$$

Let ,

$$\begin{array}{l}f(x)= x^{3}\end{array}$$

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

$$\begin{array}{l}g(t)= 6t^{2}+5\end{array}$$
at

(i) t = 0

(ii) t = 2

Solution:

The given function is

$$\begin{array}{l}g(t)= 6t^{2}+5\end{array}$$

(i) At t = 0,

$$\begin{array}{l}g(0)= 6(0)^{2}+5 \end{array}$$

= 5

(ii) At t = 2,

$$\begin{array}{l}g(2)= 6(2)^{2}+5 \end{array}$$

= 29

## Types of Functions

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

• One – one function (Injective function)
• Many – one function
• Onto – function (Surjective Function)
• Into – function
• Polynomial function
• Linear Function
• Identical 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

Practice: Find the missing equations from the above graphs.

## Composite and Periodic Functions

### 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) = x2 + 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.

Lets say we have function,

$$\begin{array}{l}f(x)=\left\{\begin{matrix} x^2 & ; & x\geq 0\\ -x^2 & ; & x<0 \end{matrix}\right.\end{array}$$

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)=

$$\begin{array}{l}h_{0}+h_{1}a+…..+h_{n}a^{n}\end{array}$$
, where n
$$\begin{array}{l}\epsilon\end{array}$$
N, and
$$\begin{array}{l}h_{0}+h_{1}+…..+h_{n}\end{array}$$
$$\begin{array}{l}\epsilon\end{array}$$
P, for each a
$$\begin{array}{l}\epsilon\end{array}$$
P, is called polynomial 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 one, it’s called a linear function. Example: b = a+1.
• Graph type: Always a straight line.

So, a polynomial function can be expressed as :

$$\begin{array}{l}f(x)= a_{n}x^{n}+a_{n-1}x^{n-1}+…..+a_{1}x^{1}+a_{0}\end{array}$$

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:

1. The polynomial function is called a Constant function if the degree is zero.
2. The polynomial function is called a Linear if the degree is one.
3. The polynomial function is Quadratic if the degree is two.
4. The polynomial function is Cubic if the degree is three.

### Linear Function

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

$$\begin{array}{l}b\in R\end{array}$$
& a ≠ 0 are called as linear functions. The graph will be a straight line. In other words, a 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 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)

$$\begin{array}{l}f\left( x \right)\text{ }=\text{ }g\left( x \right)\forall x\in {{D}_{f}}\And \text{ }{{D}_{g}}\end{array}$$

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

$$\begin{array}{l}\frac{1}{{}^{1}/{}_{x}}\end{array}$$

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

But g(x) =

$$\begin{array}{l}\frac{1}{{}^{1}/{}_{x}}\end{array}$$
is not defined of x = 0

Hence it is identical for

$$\begin{array}{l}x ~\in R\text{ }\text{ }\left\{ 0 \right\}\end{array}$$

All functions in the form of y = ax2 + bx + c where a, b,

$$\begin{array}{l}c\in R\end{array}$$
, a ≠ 0 will be known as Quadratic function. The graph will be parabolic.

At

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

In simpler terms,

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

F(x) = ax2 + bx + c, and a is not equal to zero.

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

Example, f(x) = 2x2 + x – 1 at x = 2

If x = 2, f(2) = 2.22 + 2 – 1 = 9

For Example: y = x2 + 1

### Rational Function

These are the real functions of the type

$$\begin{array}{l}\frac{f(a)}{g(a)}\end{array}$$
where 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) =
$$\begin{array}{l}\frac{f(a+1)}{g(a+2)}\end{array}$$
,
$$\begin{array}{l}\forall a\epsilon\end{array}$$
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,

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

### Cubic Function

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

F(x) = ax3 + bx2 + cx + d and a is not equal to zero.

In other words, any function in the form of f(x) = ax3 + bx2 + cx + d, where a, b, c,

$$\begin{array}{l}d\in R\end{array}$$
& a ≠ 0

For example: y = x3

Domain

$$\begin{array}{l}\in\end{array}$$
R

Range

$$\begin{array}{l}\in\end{array}$$
R

### Modulus Function

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

$$\begin{array}{l}f(a)=\left | a \right |\end{array}$$
= a, a
$$\begin{array}{l}\geq\end{array}$$
0, and f(a) = -a if a<0

$$\begin{array}{l}\forall a\epsilon\end{array}$$
P is called the modulus function.

• Domain of f = P
• Range of f =
$$\begin{array}{l}P^{+}\cup {0}\end{array}$$

$$\begin{array}{l}y=|x|=\left\{\begin{matrix} x & x\geq 0\\ -x & x<0 \end{matrix}\right.\end{array}$$

Domain: R

Range: [0,

$$\begin{array}{l}\infty\end{array}$$
)

### Signum Function

The real function f : P → P defined by

$$\begin{array}{l}\left\{\begin{matrix}\frac{\left | f(a) \right |}{f(a)}, a\neq 0 \\ 0, a=0 \end{matrix}\right.\end{array}$$
=
$$\begin{array}{l}\left\{\begin{matrix} 1,if x>0\\ 0, if x=0\\ -1, if x<0\end{matrix}\right.\end{array}$$

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 (x21) =1

### Greatest Integer Function

The real function f : P → P defined by f (a) = [a], a

$$\begin{array}{l}\epsilon\end{array}$$
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] = 2

Domain

$$\begin{array}{l}\in\end{array}$$
R

Range

$$\begin{array}{l}\in\end{array}$$
Integers

### 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) = x2sinx

f(-x) = -x2sinx

Here, f(x) = -f(-x)

it is odd function.

Example 2:

$$\begin{array}{l}f(x)={{x}^{2}}\end{array}$$

and

$$\begin{array}{l}f(-x)={{x}^{2}}\end{array}$$

f(x)=f(-x)

It is an even function.

### 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) = x2 & g(x) = 2x

f(g(x)) = f(2x) = (2x)2 = 4x2

g(f(x)) = g(x2) = 2x2

### Constant Function

The function f : P → P defined by b = f (x) = D, a

$$\begin{array}{l}\epsilon\end{array}$$
P, where D is a constant
$$\begin{array}{l}\epsilon\end{array}$$
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) =

$$\begin{array}{l}a_{0}\end{array}$$
=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 → P defined by b = f (a) = a for each a

$$\begin{array}{l}\epsilon\end{array}$$
P is called the identity function.

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

## One-One and Onto Functions

### What do you mean by a function in Mathematics?

A relation f from set A to set B is called a function if every element of set A has one and only one image in set B.

### What do you mean by domain of a function?

The domain of a function is the set of all possible inputs for a function.

### What do you mean by range of a function?

The range of a function is the set of all possible output values.

### What do you mean by a constant function?

Constant function is a function whose output is the same for every input value. For example, f(x) = 3. Here for every value of x, output will be 3.

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