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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
Matematika Terapan 12014/2013 M. Ilyas Hadikusuma, M.Eng
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.
Relation and FunctionRelation and Function
Objectives
� Definition of Function
� Function Properties
� Composition of Function
Matematika Terapan 12014/2013 M. Ilyas Hadikusuma, M.Eng
� Composition of Function
� Function Inversion
Relation and FunctionRelation and Function
Definition of Function
As mention above, function is a relation. However, this definition cannot be interchanged because the function has an
Matematika Terapan 12014/2013 M. Ilyas Hadikusuma, M.Eng
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.
Relation and FunctionRelation and Function
Matematika Terapan 12014/2013 M. Ilyas Hadikusuma, M.Eng
Some Functions
Relation and FunctionRelation and Function
Matematika Terapan 12014/2013 M. Ilyas Hadikusuma, M.Eng
Not a Function
Relation and FunctionRelation and Function
A function A to B can be written as :
f : A � B
A and B is called as Domain and Codomain respectively.
Matematika Terapan 12014/2013 M. Ilyas Hadikusuma, M.Eng
respectively.
We also can write it as
f(a) = b,
where a∈A and b∈B
Relation and FunctionRelation and Function
Denote all elements into a function form!!!
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.4
Matematika Terapan 12014/2013 M. Ilyas Hadikusuma, M.Eng
.a
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.&
.r
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Relation and FunctionRelation and Function
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.
Matematika Terapan 12014/2013 M. Ilyas Hadikusuma, M.Eng
x∈Z. Draw the graph as well.
Relation and FunctionRelation and Function
Find the domains and codomains from the graph f(x) = x2+2 below !!
Matematika Terapan 12014/2013 M. Ilyas Hadikusuma, M.Eng
Relation and FunctionRelation and Function
Function Properties
There are three basic type of function, those are :
1. One-to-one function (injective)
Matematika Terapan 12014/2013 M. Ilyas Hadikusuma, M.Eng
1. One-to-one function (injective)
2. Onto function (surjective)
3. Bijective function (both one-to-one and onto)
Relation and FunctionRelation and Function
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.
Matematika Terapan 12014/2013 M. Ilyas Hadikusuma, M.Eng
have distinct match in the codomain B.
Relation and FunctionRelation and Function
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.
Matematika Terapan 12014/2013 M. Ilyas Hadikusuma, M.Eng
element of domain A.
Relation and FunctionRelation and Function
Bijective Function
Bijective can be called correspondence one to
one. It can be inverted.
Matematika Terapan 12014/2013 M. Ilyas Hadikusuma, M.Eng
Relation and FunctionRelation and Function
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
Matematika Terapan 12014/2013 M. Ilyas Hadikusuma, M.Eng
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)).
Relation and FunctionRelation and Function
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
Matematika Terapan 12014/2013 M. Ilyas Hadikusuma, M.Eng
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.
Relation and FunctionRelation and Function
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
Matematika Terapan 12014/2013 M. Ilyas Hadikusuma, M.Eng
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?
Relation and FunctionRelation and Function
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
Matematika Terapan 12014/2013 M. Ilyas Hadikusuma, M.Eng
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.
Relation and FunctionRelation and Function
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?
Matematika Terapan 12014/2013 M. Ilyas Hadikusuma, M.Eng
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.