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CHAPTER 1
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Relations andFunctions
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Cross Product
Let A and B be sets.
The cross product of A and B, denoted by
A x B is the set of all ordered pairs (x, y)such that x A and y B.
Notation:
A x B = {(x, y)` x A and y B}
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Examples:
1.
2.
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Relations
A relation from A to B is anynonempty subset of A x B .
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1. Any set of points in the rectangularCartesian coordinates is a relation.
Examples:
A = {(1,2),(3,5),(4,6),(8,9)}
B = {(1,0),(2,0),(1,5)}
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2.Equation and inequality are relations.
3. Tables, graphs and mappings also
represent relations.
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FunctionsFunctions
Definition:
A function f from A to B is a relation from A to B
where to each a A, there corresponds exactly
one b B.
Alternative Definition:A function is a set of ordered pairs in which no two
ordered pairs have the same first component.
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1.
Examples: Identify which of the following
are functions.
A = {(1,2),(3,5),(4,5),(8,9)}
B = {(1,0),(2,0),(1,5)}
C = {(1,3),(1,5),(2,5),(3,1),(4,-1)}
D = {(-1,2),(1,3),(2,3)}
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2.
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V
ertical Line TestGiven the graph of a relation, if
any vertical line constructedintersects (or passes) the graphin at most one point, then the
relation described by the graphis that of a function.
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Domain and Range
Consider the function
The set of all x such that isthe domain of f, denoted by
The set of all y such that isthe range of f, denoted by
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The domain is also defined as the set of
all permissible values of x andrange as the set of all correspondingvalues of y.
Domain and Range
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Examples:
1. A = {(1,2),(3,5),(4,5),(8,9)}
Domain: {1, 3, 4, 8} Range:{2, 5, 9}
2.
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Functions
Two main types of functions:
Algebraic functions ² those functions that can
be obtained by a finite combination of constants
and variables together with the four basic
operations, exponentiation, or root extractions.
Transcendental functions ² those that are not
algebraic.
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AlgebraicFunctions
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POLYNOMIAL FUNCTIONSGeneral Form:
where n is a non-negative integer and
012
23
32
21
1 a xa xa xa xa xa xa x P nn
nn
nn !
-
a0,a1,a2,..., an R
The domain of any polynomial function is the
set of all real numbers.
If , the polynomial function P is said
to be of degree n.
0{na
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Standard Form:
, where b is a real number.
Constant Functions
y f x b! !
Domain:Range:
f D !
_ a f R b!
Graph:Horizontal line passing through (0 ,b).
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Constant Functions
Examples
Find the domain, range and sketch the
Graph of the following:
1. 2 y f x! !
2. 3 y g x! !
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Linear Functions
Standard Form:
where m and b are any real numbers.
Domain:
Range:
or
or
Graph: Straight line
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Examples of Linear Function
4
523
!
x ) x( h.
2. ( ) 3 g x x!
421 ! x ) x( f .
Non-examples
42 . ( )
2 5i x
x!
11. ( )n x
x!
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How do we sketch the graph of a
linear function?
It·s enough to sketch a line using any 2
points.
We can use
y-intercept: the value of y when x=0 x-intercept: the value of x when y=0
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Note:
y-intercept: the value of y when x=0
x-intercept: the value of x when y=0
Thus , if , then
y-intercept: -4
x-intercept: 2
( ) 2 4 y g x x! !
y-intercept: y=-4
x-intercept: x=2
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Illustrations:421 ! x ) x( f .
y-intercept: -4
x-intercept: 2
4
,0
0,2
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Illustrations:y-intercept: 4
x-intercept: 4 x ) x( g . ! 42
4,0
0,4
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Illustrations:y-intercept: ?
x-intercept: ?
¹ º ¸©
ª¨
4
50 ,
¹ º ¸©
ª¨ 0
2
5 ,
4
523
!
x ) x( h.
int :
5 50, int :4 4
int :
0, 0 2 5
2 5
5 5int :2 2
f or y
when x y y
f or x
when y x
x
x x
! !
! !
!
!
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Illustrations:
4. ( ) g x x!
2,2
1,1
0,0