matematika terapan week 5 [compatibility mode]

TIF 21101

APPLIED MATH 1

(MATEMATIKA TERAPAN 1)

Matematika Terapan 12014/2013 M. Ilyas Hadikusuma, M.Eng

Week 5

Relation and Function II

Relation and FunctionRelation and Function

OverviewIn mathematics, function is a relation. A function establishes or expresses the “relation”-ship between objects. In computer systems, for instance, the input is fed to the system in form of data or objects and the system generates the


data or objects and the system generates the output that will be the function of input. So, in other words, function is the mapping or transformation of objects from one form to other.

In this section we will concentrate our discussion on function and its classifications.


Objectives

� Definition of Function

� Function Properties

� Composition of Function


� Composition of Function

� Function Inversion


Definition of Function

As mention above, function is a relation. However, this definition cannot be interchanged because the function has an


interchanged because the function has an unique relationship.

Let A and B is non-empty sets. A relation from A to B is said as a function if all element of A has only “one connection” to B.



Some Functions



Not a Function


A function A to B can be written as :

f : A � B

A and B is called as Domain and Codomain respectively.


respectively.

We also can write it as

f(a) = b,

where a∈A and b∈B


Denote all elements into a function form!!!

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.s

.4


.a

.7

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.r

.10


Frequently, function is expressed in mathematical formulas. Example:

Find the codomain of f(x) = 2x+3 for 1≤x≤ 4, x∈Z. Draw the graph as well.


x∈Z. Draw the graph as well.


Find the domains and codomains from the graph f(x) = x2+2 below !!



Function Properties

There are three basic type of function, those are :

1. One-to-one function (injective)


1. One-to-one function (injective)

2. Onto function (surjective)

3. Bijective function (both one-to-one and onto)


Injective Function

A function f: A � B is said to be one-to-one

(written 1-1) if different elements in the domain A

have distinct match in the codomain B.


have distinct match in the codomain B.


Surjective Function

A function f: A � B is said to be an onto function if

each element of codomain B is the image of some

element of domain A.


element of domain A.


Bijective Function

Bijective can be called correspondence one to

one. It can be inverted.



Composition of FunctionLet g be a function from the set A to the set B and let f be a function from the set B to the set C. The composition of the functions f and g, denoted by f o g, is defined by


f o g, is defined by

(f o g)(a) = f(g(a)).

Therefore, to find (f o g)(a) we first apply the function g to a to obtain g(a) and then we apply the function f to the result g(a) to obtain (f o g)(a) = f(g(a)).


Ex.

Let g be the function from the set (a, b, c} to itself such that g(a) = b, g(b) = c, and g(c) = a. Let f be the function from the set {a, b, c} to the set {1, 2, 3} such that f(a) = 3, f(b) = 2, and f(c) = 1. What is the


such that f(a) = 3, f(b) = 2, and f(c) = 1. What is the composition of f and g?

Solution :

The composition f o g is defined by (f o g)(a) = f(g(a)) = f(b) = 2, (f o g) (b) = f(g(b)) = f(c) = 1, and (f o g)(c) = f(g(c)) = f(a) = 3.


Exercise

Let f and g be the functions from the set of integers

to the set of integers defined by f(x) = 2x + 3 and

g(x) = 3x + 2. For x = 2 and -2, what is the


g(x) = 3x + 2. For x = 2 and -2, what is the

composition of f and g? What is the composition of

g and f?


Function Inversion

Let f be a one-to-one correspondence from the set

A to the set B. The inverse function of f is the

function that assigns to an element b belonging to


function that assigns to an element b belonging to

B the unique element a in A such that f(a) = b.

The inverse function of f is denoted by f-1. Hence,

f-1(b)= a when f(a) = b.


Ex.

Let f be the function from {a, b, c} to {1, 2, 3} such that f(a) = 2, f(b) = 3, and f(c) = 1. Is the f invertible? If it is, what is its inverse?


Solution:

The function f is invertible because it is a one-to-one correspondence. The invers function f reverses the correspondence given by f, so

f-1(1) = c, f -1(2) = a, and f-1(3)=b.


For mathematics formulas, to find invers of f, we can exchange the variable x with y.

Ex.


matematika terapan week 5 [compatibility mode]

Education

function form

function of input

function iirelation

bijective function

function surjective3

ilyas hadikusuma

onetoone function injective2

basic type of function