Matematika terapan week 5

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
Overview
In 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
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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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.

Let 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

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