Functions In Math

Mathematics lesson, subject of functions. Concept of domain, codomain and range in functions. What is function. Relations and functions. The definition of function.


Definition Of Function

Let we take non-empty set of A and B.

The relation that mapping up least and most 1 time each elements of set A to elements of set B is called function.

There is two important rule in a function.

1-Each element of set A (is domain) must maps to with an element of set B(is codomain).

2- An element of set A (is domain) must maps to with an element of set B (is codomain) most 1 time.

 A function from A to B is shown as

f: A → B

Or

Functions-k1-i1              


Domain set of A Function

In a function from A to B the A set is called set of domain or set of value.


Codomain set of A function

In a function from A to B the B set is called the set of codomain or the set of definition.


The Range Set of A Function

In the B set, the elements that matches with the elements of set A is create a set. This set is called range set or image set.


The Relationship of A function

A relation that pairs the elements of set A to elements of set B is called the relationship of function.

Relationship is shown with arrow or mathematical formula.


Example:

A f(x) function is given in the following.

f(x) = 3x + 1


The name of this function is f.

The input of this function is "x".

The output of this function is "3x + 1".

There is the relationship "x → 3x + 1" in this function.


The function output for x = 2,

f(2 ) = 3*2 + 1 = 7 dir.

The function output for x = 5

f(5) = 3*5 + 1 = 16

The function output for x = 8

f(8) = 3*8 + 1 = 25 


The functions can be defined in various form simple or mixed.


Defining Functions with Finite Sets

It is seen a function from A to B in the following.

Functions-k1-i2


Example

In the following in the example is shown that the elements of the set A to pair with the elements of set B.

It is not a function. Because, all elements (c, d) of the set A is not mapping with an element of set B.


Functions-k1-i3


Example:

In the following in the example is shown that the elements of the set A to pair with the elements of set B.

It is not a function. Because, an element(is d) of the set A matched with element of set B more than once.

Functions-k1-i4b


Example:

It is a function. Because, each elements of the set A matched with element of set B at least and most 1 times.

Functions-k1-i5



Example:

A and B sets is given, 

A = {2, 3, 5, 6}

B = Whole Numbers

f(x) : A → B

f(x) = 5x + 2

Find the image(range) set of the function.


Solution:

The elements of the range set can shown in the following form

R = {b1, b2, b3, b4}


b1 = 5.2 + 2 = 12

b2 = 5.3 + 2 = 17

b3 = 5.5+2 = 27

b4 = 5.6 + 2 = 32

R = {12, 17, 27, 32}


Also, the range set (R) can also shown in "f(A)" form.

R = f(A)

Example:

f: A → B

f(x) = 3x + 6 

f(A) = {12, 18, 24, 33}

Find elements of the set A


Solution:

Elements of the set A must matches up at most and least 1 time with elements of the set B.


A = {a1, a2, a3, a4}

3*a1 + 6 = 12

3*a1 = 6

a1 = 2


3*a2 + 6 = 18

3*a2 = 12

a2 = 4



3*a3 + 6 = 24

3*a3 = 18

a3 = 6


3*a4 + 6 = 33

3*a4 = 27

a4 = 9

Let we write the set A.

A = {2, 4, 6, 9}


Compound Functions



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11/08/2018

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