lecture 1.1 real no , relation & functions
TRANSCRIPT
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1-1: Real Numbers,
Relations & Functions
Learning Goals:Rational & irrationalnumbers
coordinate system in aplane.concept of relations &
functions
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Definition
The Real Number System:
Real Numbers
Rational
Numbers
Irrational
Numbers
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Definition
Rational Numbers can beexpressed in the form
r
s
where r and s are integersand 0s { . Rational numbers
can be written as a decimalthat either terminates or
repeats in a pattern.
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Example
Rational Numbers:
1
4
r
0s {
5
-3
2
3
7
3
5560
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Important Idea
0 1-4 57
2
A rational number is anumber that can befound on a number line.
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Definition
Irrational Numbers are
numbers that cannot be
expressed in the form
and cannot be found on anumber line.
r
s
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Example
Irrational Numbers:
2
5
95
T
e
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Important Idea
Any number that cannot bewritten as a decimal thateither terminates or repeatsin a pattern is an irrational
number. Square roots ofnumbers that are not perfect
squares are always irrational.
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Important Idea
Numbers such as arecalled irrational numbers
and are always betweentwo other numbers.
2
1 2
1.51.4
2
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...If is always between
two numbers, whereexactly is it?
2
Could it be that it doesntexist ???
Important Idea
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Try This
True or False:
9 is an irrational number.
False
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Definition
CartesianCoordinate
System:
III
III IV
quadrants
axes
origin
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ExamplePlot thefollowingordered pairs:
(0,5)
(2,4)
(3,-5)
(-4,-4)
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Definition
The Domain is all thepossible values thatx canhave. The Range is all thepossible values thaty can
have.f(x) is another name for y.
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Example
The table shows the height andarm span in cm. of six membersof the cheerleading squad:
Height 178 189 158 169 195 188
Arm
Span
180 180 163 169 189 182
Write the relations domain andrange using set notation.
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Example
Given the relation{(0,1),(1,2), (-1,2),(2,5),(-
2,5)(3,10),(-3,10)}, state itsdomain & range, create ascatter plot and find a rulethat relates the value of thefirst coordinate to the value
of the second coordinate.
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Try This
Given the relation{(0,0),(1,1),(1,-1),(4,2),(4,-
2)(9,3),(9,-3)}, state itsdomain & range, create ascatter plot and find a rulethat relates the value of thefirst coordinate to the value
of the second coordinate.
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Solution
Domain={0,1,4,9}Range={-3,-2,-1,0,1,2,3}
Rule: 2x y! or y x! s
Scatter Plot:Lists:
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Definition
Afunction is a set of orderedpairs such that the firstcoordinate denotes the input,the second coordinatedenotes the output that is
obtained from th
e rule of th
efunction, and each inputcorresponds to one and only
one output.
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Important IdeaA function is likea meat grinder.
You putsomething in(Domain), turnthe crank(implement the
Rule), andcollect
the output (Range).
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Example
In the following sets of orderedpairs, the first coordinate
represents an input and the
second coordinate representsthe corresponding output.Explain why each set is, or is
not, a function.a) {(0,1),(-1,3),(2,4),(2,-1)}
b) {(0,1),(-1,3),(2,4),(3,-1)}
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Try This
In the following set of orderedpairs, the first coordinate
represents an input and the
second coordinate representsthe corresponding output.Explain why each set is, or is
not, a function.a) {(0,1),(1,3),(2,4),(3,5)}
b) {(0,1),(2,3),(2,4),(3,5)}
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Example
These areexamples of
functions. Eachinput musthave
one and only
one output.{(0,1),(1,3),(2,4),(3,5)}
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These areexamples ofrelations which
are notfunctions. Eachinputhas morethan oneoutput.
{(0,1),(1,3),(1,4),(3,5)}
Example
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Definition
Function Notation: Instead of2 3y x! , we write
2( ) 3. f x x! If the input is 2,
then t
he output is 7, t
hat is :( 2 ) 7 .f ! The rule is: 2 3x
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Try ThisFor the function: ( ) f x x!
What is ,(4)f (9)f ( 4)?f &
What is the domain (inputs) andrange (outputs)?
(4) 2f !(9) 3f !
( 4)f is undefined
D= { }x ouR={ 0}y u