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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 + ac a(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 DomainRange 3π 11- 2 1.6182.718

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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 its heart. The relation is the time of the day and the intensity of the sun light. The relation is a number and its 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 y1y2y1y2 RANGE R NOT A FUNCTION

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y RANGE f FUNCTION CONCEPT x1x1 DOMAIN x2x2

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Examples Decide if the following relations are functions. X Y 1 2 -5 7 -1 2 3 X Y 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 SymbolicSymbolic X Y 1 2 5 10 -1 -2 3 6 GraphicalGraphical NumericNumeric VerbalVerbal The cost is twice the original amount.

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Example Penneys is having a sale on coats. The coat is marked down 37% from its 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 Notation The Symbolic Form A truly excellent notation. It is concise and useful.

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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

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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.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 Symbolically Graphically Numerically Verbally

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Definition of a Function A function is a set of ordered pairs in which no two ordered pairs of the form (x, y) have the same x-value with different y-values.

Definition of a Function A function is a set of ordered pairs in which no two ordered pairs of the form (x, y) have the same x-value with different y-values.

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