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Chapter Four
Functions
Section 4.1
A function from a set D to set R (f: DR) is a relation
from D to R such that each x in D is related to one
and only one y in R. D is called the domain of the
function and R is called the range of the function.
 Note: every function is a relation but not every
relation is a function.
 Ex:
The cost of gold is a function of its weight.

Functions

Functions are represented by cloud
diagrams.
X
f(x)
t
f(t)
Functions
Ex1:
If f(x)=x+2 is used as a formula that defines a
function from {-1,0,1,2} to {1,2,3,4}, what
relation defines f??

Solution:
f = {(-1,1),(0,2),(1,3),(2,4)}
Functions
Ex2:
The relation R ={(a,2),(a,3),(b,4),(c,5)} is not a
function from D={a,b,c} to R={2,3,4,5}, why?

Functions
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Ex3:
The relation {(a,3),(c,2)} from the domain
{a,b,c} to the range {2,3,4,5} is not a function.
why?
Functions
To determine if a relation is a function:
1- each element in the domain is related to an
element in the range.
2- no element in the domain is related to more
than one element in the range.
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Functions
A function can have 2 or 3 variables or more.
f(x,y)= xy+x^2
G(x,y,z)= x+2y+z
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One to one functions
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A function is one to one (or injection) if
different elements of the domain are related
to different elements of the range.
Ex:
1
a
2
3
b
c
4
d
Ex:
Is the following one-to-one? Why?
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1
a
2
3
b
c
4
d
Onto functions

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A function is called onto (surjection) if each
element of the range is related to at least one
element in the domain.
Ex:
1
a
2
3
b
c
d
Bijection functions


A function is one to one and onto (bijection)
or one to one correspondence if each
element in the range is related to one and
only one element of the domain.
Ex:
4
1
2
a
b
3
c
d
Functions
Ex4:
Determine whether each if the following is a function, if so is it
onto? Is it one to one?
1- D= {a,b,c,d} R={1,2,3,4}
F= { (a,1),(a,2),(b,1), (c,2), (d,3)}
2- D= {-2,-1,0,1,2} R= {0,1,4}
F(x)=x^2
3- D= {-2,-1,0,1,2} R= {0,1,2,3,4}
F(x)=x^2
4- D= {0,1,2} R= {0,1,4}
F(x)=x^2

The image of a function

The set of range values actually related to
some domain elements is called the image of
a function.
Sequences, n-tuples and sums
A function whose domain is a set of
consecutive integers is called a sequence.
 Ex:
If s (i)=i for each i>=0, then s is a sequence.
Si is the ith term of the sequence.

Sequences, n-tuples and sums
Ex5:
Write the third term of the sequence
Si= i(i-1)+1 for i>=1

Solution: s3=3(2)+1=7
Sequences, n-tuples and sums
Ex6:
Write the first 5 terms for the sequence si=i^2+2, for
i>=0

Solution:
S0= 2
S1= 3
S2=6
S3=11
S4=18
Sequences, n-tuples and sums
Ex7:
Find a formula for the ith term of the sequence
1,4,9,16,25. For what values of i is your
formula valid?

Solution: si=i^2, i>=1
Sequences, n-tuples and sums


(1,2,4,9,16) is a 5-tuple.
Order is important.
Summing Finite Sequences
∑ ai (i=m, n) = am + am+1 + am+2 +…..+an
 Ex8:
Find ∑ i^2 (i=1, 3).
Solution:
1+4+9=14
Ex9:
Find ∑ 2j-1 (j=0, 4).
Solution:
-1+1+3+5+7=15

Gauss’s Formula

∑ i (i=1, n) = n(n+1)/2
Proof given in class.
Summing Finite Sequences
Theorem:
- ∑(ai+bi) (i=m,n) = ∑ ai (i=m,n) + ∑ bi (i=m,n)
- ∑cai (i=m,n)= c ∑ai (i=m,n)
- ∑c (i=m,n)= c(n-m+1)
 Theorem:
The sum of the arithmetic series ∑(a*i+b)
(i=1,n) = an(n+1)/2+nb
*proof given in class

Summing Finite Sequences
Ex:
solve
∑(3i-1)(i=1,10)

Section 4.2
Cartesian graphs
The Cartesian graph of a relation R consists of all points (x,y) in
the plane such that x is related to y by R ( that is, (x,y) ε R or
xRy).
Ex: Graph of
- X2+y2 =1
- Y=X2
- Y=X3
- Y=x
- Y=2x
Graphs drawn in class

Functions
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Vertical lines are used to determine whether a certain relation is
a function
Horizontal lines are used to determine whether a function is one
to one or onto.
If all vertical lines cross the graph exactly once over a certain
domain then the graph is a function on that domain
If the horizontal lines cross the function’s graph more than once
it is not one to one.
If there are horizontal lines that do not cross the function’s
graph on a certain range, the function is not onto on that range.
Composition and Inverse


The composition of function f with a function
g is the relation f g (f composed with g) that
contains the pair (x,y) if and only if y=f(g(x)) (f
of g of x)
The image of g must be a subset of the
domain of f in order for f g to be defined
Composition and Inverse
Find f(g(x))
1- f = { (1,-1), (2,-2),(3,-3),(4,-4),(5,-5)}
g= { (1,3), (2,4), (3,5)}
2- f(y) = y3+1
g(x)= x 1/3
3- f(x)= x3+1
g(x)= (x-1)1/3

Composition and Inverse
Ex:
454g in 1 lb
16 oz in 1lb
P(x)= x/16
G(x)= 454x
Find the equation that converts from ounces
(oz) to pounds (lb)

Inverses
Whenever f and g are two functions such that
f(g(x))=x and g(f(y))=y for each x in the domain of g
and each y in the domain of f, we say that f and g are
inverses of each other, f is the inverse of g and g is
the inverse of f
 Theorem:
A function g from D to R has an inverse if and only if
g is one to one. The domain of g-1 is the image of g.

Inverse
Ex:
G(x) = x3 -1
Find the inverse and its domain

Inverse
Theorem:
If g is one to one, the points on the graph of g-1 may
be obtained by interchanging the x and y coordinates
of the points on the graph of g.
Ex:
Find the inverse of y=x3 and its graph
Ex:
F(x) = 2x is to one from R to R+
Find the inverse and its graph
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Section 4.3 ( Growth Rate of Functions)
Algorithm smallest
Input: a list of numbers in any order
Output: a list of the same numbers with the smallest first
Steps:
For i from 2 to n do:
If (the number in position i < the number in position 1)
Then exchange both numbers
Example: apply algorithm on 5,2,1,3 and on 5,4,3,2 for the worst
case
- Find the total time taken by the algorithm
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Quadratic time algorithms
Selection Sort
For(i=0; i<=n-2; i++)
{
L= min-position(iteration,n-1);
Exchange (list [iteration], list [L]);
}
- Apply on 8,2,4,0,1,3
- Find the total time taken by the algorithm
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Growth Rate of Functions
We say that f is a big O of g, or f(x) = O(g(x)),
If there is a constant c>0 and a number N such
that for all x>N, f(x) <= c.g(x)
Ex:
f(x)= x2
g(x)= x3
Show that f(x) = O(g(x))
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Growth Rate of Functions
Consider
F(n) = 3n2+2n
G(n)= n2
Show that f(n)= O(g(n))
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Growth Rate of Functions
If lim x∞ g(x)/f(x) = ∞
Then g(x) is not O(f(x))
 If lim x∞ g(x)/f(x) = 0
Then g(x)=O(f(x)), but not vise versa
 If lim x∞ g(x)/f(x) = c, where c is constant ≠ 0
Then f(x)=O(g(x)) and g(x)=O(f(x))
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Growth Rate of Functions
Ex:
f(x)= 10x
G(x)= x2
H(x)=x3
Is f(x) = O(g(x))?
Is G(x)=O(h(x))?
Is G(x)=O(f(x))?
Growth Rate of Functions
 Ex:
F(x)=2x
G(x)=4x
Growth Rate of Functions
Theorem
For any real number r, x= O(2x) and
1- 2x is not O(xr)
2- logx = O(xr)
3- xr is not O(logx)

Growth Rate of Functions
Ex:
F(x)= xx
G(x)= x2x
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Growth Rate of Functions
1- constants
2- log(log(x))
3- log(x)
4- (log(x))n
5- (x)1/k
6- x
7- x2
8- xn
9- 2x
10- x!
Theorem
F(x) is of order g(x) means that f(x)= O(g(x))
and g(x)= O(f(x))
Ex:
Which of the following functions are O(x2) and which
of them are of order x2?
- f(x) = 2x(x)1/2
- f(x)= x2+(x)1/2
- F(x)=x3+(x)1/2
- F(x)=(x3+1)/x
