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Functions and Graphs
The Mathematics of Relations
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Definition of a Relation
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Relation
(1)32 mpg(2)8 mpg(3)16 mpg
(A)
(C)
(B)
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Domain and Range
• The values that make up the set of independent values are the domain
• The values that make up the set of dependent values are the range.
• State the domain and range from the 4 examples of relations given.
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Quick Side Trip Into the Set of Real Numbers
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The Set of Real Numbers
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Ponder
• To what set does the sum of a rational and irrational number belong?
• How many irrational numbers can you generate for each rational number using this fact?
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Properties of Real Numbers• Transitive:
If a = b and b = c then a = c• Identity:
a + 0 = a, a • 1 = a• Commutative:
a + b = b + a, a • b = b • a• Associative:
(a + b) + c = a + (b + c)(a • b) • c = a • (b • c)
• Distributive: a(b + c) = ab + aca(b - c) = ab - ac
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Definition of Absolute Value
if a is positive
if a is negative
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The Real Number Line
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End of Side Trip Into the Set of Real Numbers
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Definition of a Relation
• A Relation maps a value from the domain to the range. A Relation is a set of ordered pairs.
• The most common types of relations in algebra map subsets of real numbers to other subsets of real numbers.
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Example
Domain Range
3 π
11 - 2
1.618 2.718
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Define the Set of Values that Make Up the Domain and Range.
• The relation is the year and the cost of a first class stamp.
• The relation is the weight of an animal and the beats per minute of it’s heart.
• The relation is the time of the day and the intensity of the sun light.
• The relation is a number and it’s square.
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Definition of a Function
• If a relation has the additional characteristic that each element of the domain is mapped to one and only one element of the range then we call the relation a Function.
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Definition of a Function• If we think of the domain as the set of all
gas pumps and the range the set of cars, then a function is a monogamous relationship from the domain to the range. Each gas pump gets used by one car.
• You cannot put gas in 2 cars as the same time with one pump. (Well not with out current pump design )
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x
DOMAIN
y
RANGE
f
FUNCTION CONCEPT
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x
DOMAIN
y1
y2
RANGE
R
NOT A FUNCTION
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y
RANGE
f
FUNCTION CONCEPT
x1
DOMAIN
x2
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Examples
• Decide if the following relations are functions.
X Y
1 2
-5 7
-1 2
3 3
X Y
1 1
-5 1
-1 1
3 1
X Y
1 2
1 7
1 2
1 3
X Y
1 π
π 1 -1 5
π 3
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Ponder
• Is 0 an even number?• Is the empty set a function?
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Ways to Represent a Function• Symbolic
x,y y 2x or
y 2x
X Y
1 2
5 10
-1 -2
3 6
• Graphical
• Numeric
• VerbalThe cost is twice the original amount.
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Example
• Penney’s is having a sale on coats. The coat is marked down 37% from it’s original price at the cash register.
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• If you chose a coat that originally costs $85.99, what will the sale price be? What amount will you pay in total for the coat (Assume you bought it in California.)
• Is this a function? What is the domain and range? Give the symbolic form of the function. If you chose a coat that costs $C, what will be the amount $A that you pay for it?
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Function NotationThe Symbolic Form
• A truly excellent notation. It is concise and useful.
y f x
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y f x • Output Value• Member of the Range• Dependent Variable
These are all equivalent names for the y.
• Input Value• Member of the Domain• Independent Variable
These are all equivalent names for the x.
Name of the function
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Example of Function Notation
• The f notation
f x x 1
f 2 2 1
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Graphical Representation• Graphical representation of functions
have the advantage of conveying lots of information in a compact form. There are many types and styles of graphs but in algebra we concentrate on graphs in the rectangular (Cartesian) coordinate system.
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Average National Price of Gasoline
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Graphs and Functions
Domain
Range
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CBR
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Vertical Line Test for Functions
• If a vertical line intersects a graph once and only once for each element of the domain, then the graph is a function.
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Determine the Domain and Range for Each Function
From Their Graph
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Big Deal!
• A point is in the set of ordered pairs that make up the function if and only if the point is on the graph of the function.
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Numeric
• Tables of points are the most common way of representing a function numerically
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Verbal
• Describing the relation in words. We did this with the opening examples.
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Key Points
• Definition of a function• Ways to represent a function
SymbolicallyGraphicallyNumericallyVerbally