01 functions
TRANSCRIPT
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Unit 8
Functions & GraphsF2N
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Definition of a Relation
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http://../Bahan%20Ceramah/Kursus%20MaSTT%20PLGSK%2017%20Nov%2009/4squares.ppthttp://../Bahan%20Ceramah/Kursus%20MaSTT%20PLGSK%2017%20Nov%2009/4squares.ppt -
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Definition of a Relation
A Relationmaps values from one subsetto the of values another subset. ARelationis a set of ordered pairs.
The most common types of relations inalgebra map subsets of real numbers toother subsets of real numbers.
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Mapping diagram illustrates how each member of
the domain is related with each member of the range
x y
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5
7
-91
2
Example: Draw a mapping for the following.(5, 1), (7, 2), (4, -9), (0, 2)
(Note: First list values of x and y once each, inorder.)
How to show relation?
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5
6
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2
3
4
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The Rule is ADD 4 to the domain
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Ahmed
Peter
Ali
Jaweria
Hamad
Paris
London
Dubai
New York
Cyprus
Has Visited
There are MANYarrows from each person and each place is related to MANYPeople. It is a MANY to MANYrelation.
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Bilal
Peter
Salma
Alaa
George
Aziz
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Person Has A Mass of Kg
In this case each person has only one mass, yet several people have the sameMass. This is a MANY to ONErelationship
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Is the length of
14
30
Pen
Pencil
Ruler
Needle
Stick
cm object
Here one amount is the length of many objects.This is a ONE to MANYrelationship
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FUNCTIONS
Many to One Relationship
One to One Relationship
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Only involve 2 kinds of relationship:-
Link
http://../Bahan%20Ceramah/Kursus%20MaSTT%20PLGSK%2017%20Nov%2009/1%5b1%5d._brainteasers.ppthttp://../Bahan%20Ceramah/Kursus%20MaSTT%20PLGSK%2017%20Nov%2009/1%5b1%5d._brainteasers.ppt -
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Domain Co-domain
0
1
2
3
4
12
345678
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Image Set (Range)
x2x+1A B
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:x
x2
4
x x2 4
The upper function is read as follows:-
Function fsuch that xis mapped onto x2+4
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Lets look at some functionType questions
Iffxx2 4andgx1-x2
Findf2
Findg3
x x2 42 2 = 8 gx 1 -x2
3 3
= -8
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Consider the functionfx 3x - 1 We can consider this as two simplerfunctions illustrated as a flow diagram
Multiply by 3 Subtract 13x 3x - 1x
Consider the functionf:x2x 52
xMultiply by 2 Add 5
2x 2x 5Square
all
2x 52
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:x3x 2 and gx :xx2Consider 2 functions
g is a composite function, where gis performed first and then fis performedon the result of g.The function fgmay be found using a flow diagram
xsquare
x2Multiply by 3
3x2Add 2
3x2 2
Thus = 3x2 2
f
f
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x
2 3x 2
3x2 2
24 14
2
f
f =15
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Consider the function x 5x - 2
3
Here is its flow diagram
x
x5x5 -2
5x - 23
Draw a new flow diagram in reverse!. Start from the right and go left
Multiply by 5 Subtract 2 Divide by three
Multiply by threeAdd twoDivide by 5
x3x3x +23x +2
5
-1x 3x 2
5And so
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Example
Find the domain and range of the relation.
{(5,12), (10, 4), (15, 6), (-2, 4), (2, 8 )}
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Example
Determine whether each relation is a function:
A) {(1,2), (3,4), (5,6), (5,8)}
B) {(1,2), (3,4), (6,5), (8,5)}
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Functions as Equations
a) b)
1. Solve for yin terms of x.
2. If two or more values of ycan be obtained
for a given x, the equation is not a function.
44222 yxyx
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Solve for y and determine if theequation is a function.
A) 2x + y = 6 B) x2 + y2 = 1
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Evaluating a Function
Common notation: f(x) = function
Evaluate the function at various values of x,represented as: f(a), f(b), etc.
Example: f(x) = 3x 7Then, f(2) =
f(3 x) =
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If f(x) = x2 2x + 7, evaluate each of the
following.
a) f(-5) b) f(x + 4) c) f(-x)
Ans: a) 42
b) x2+6x + 15
c) x2 + 2x + 7
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Determine if a relation is a functionfrom the graph?
Remember: to be a function, an x-value isassigned to ONLY one y-value .
On a graph, if the x value is paired with MOREthan one y value there would be two pointsdirectly on a vertical line.
THUS, the vertical line test! If a vertical linedrawn on any part of your graph touches morethan one point, it is NOT the graph of afunction.
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To determine Graphs ofFunctions
Step 1: Graph the relation. (Use graphingcalculator or pencil and paper.)
Step 2: Use the vertical line test to see if therelation is a function.
Vertical line test If any vertical line
passes through more than one point ofthe graph, the relation is not a function.
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(a)
(b)
(c) (d)
(a) and (c)25
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Determine if the graph is a function.
a) b)y
x
5
5
-5
-5
y
x
5
5
-5
-5
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Heres more practice.
c) d)y
5
5
-5
-5
y
x
5
5
-5
-5
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Can you identify domain & range fromthe graph?
Look horizontally. What x-values are containedin the graph? Thats your domain!
Look vertically. What y-values are contained inthe graph? Thats your range!
Write domain and range using interval or set-builder notation.
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What is the domain & range of thefunction with this graph?
) : ( , ), : ( , )
) : ( 3, ), : ( , )
) : ( 3, ), :( 3, )
) : ( , ), : ( 3, )
a Domain Rangeb Domain Range
c Domain Range
d Domain Range
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Additional Example
Graph the function.
Then estimate thedomain and range.
( ) 1f x x -
( ) 1f x x -
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Finding intercepts:
x-intercept: where the function crosses thex-axis. What is true of every point on thex-axis? The y-value is ALWAYS zero.
y-intercept: where the function crosses they-axis. What is true of every point on they-axis? The x-value is ALWAYS zero.
Can the x-intercept and the y-interceptever be the same point? YES, if thefunction crosses through the origin!
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Summary
Domain = x values
Range = y values
Use the vertical line test to verify if a graphis a function.
To evaluate means to substitute and
simplify. Intercepts where function crosses the x-
or y-axis
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