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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)xA and y B}
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Examples:
1.
2.
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Relations
A relation from A to B is any
nonempty 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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Functions
Definition:
Afunction ffrom A to B is a relation from A to B
where to each a A, there corresponds exactly
one b B.
Alternative Definition:Afunction is a set of ordered pairs in which no two
ordered pairs have the same first component.
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2.
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Vertical Line Test
Given the graph of a relation, ifany 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 ofall permissible values of x and
range 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 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 functionsthose that are not
algebraic.
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AlgebraicFunctions
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POLYNOMIAL FUNCTIONS
General Form:
where n is a non-negative integer and
012
23
32
21
1 axaxaxaxaxaxaxPn
nn
nn
n
a0,a
1,a
2,..., a
n 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.
0na
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Standard Form:
, wherebis a real number.
Constant Functions
y f x b
Domain:Range:
fD fR 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. 2y f x
2. 3y g x
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Examples of Linear Function
4
523
x)x(h.
2. ( ) 3g 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 alinear function?
Its enough to sketch a line using any 2points.
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 , theny-intercept: -4
x-intercept: 2
( ) 2 4y g x x
y-intercept: y=-4
x-intercept: x=2
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