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Functions and Graphs The Mathematics of Relations Definition of a Relation Relation (A) (B) (C) (1) 32 mpg (2) 8 mpg (3) 16 mpg 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. Quick Side Trip Into the Set of Real Numbers The Set of Real Numbers 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? 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 Definition of Absolute Value if a is positive if a is negative The Real Number Line End of Side Trip Into the Set of Real Numbers 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. Example Domain Range 3 π 11 -2 1.618 2.718 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. 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. Definition of a Function • If we think of the domain as the set of boys and the range the set of girls, then a function is a monogamous relationship from the domain to the range. Each boy gets to go out with one and only one girl. • But… It does not say anything about the girls. They get to live in Utah. FUNCTION CONCEPT f x y DOMAIN RANGE NOT A FUNCTION R x y1 y2 DOMAIN RANGE FUNCTION CONCEPT f x1 y x2 DOMAIN RANGE Examples • Decide if the following relations are functions. X Y X Y X Y X Y 1 2 -5 7 1 1 1 2 -5 1 1 7 -1 2 -1 1 1 2 3 3 1 3 3 1 1 π π 1 -1 5 Ponder • Is 0 an even number? • Is the empty set a function? Ways to Represent a Function • Symbolic x,y y 2x or y 2x • Numeric X Y 1 2 5 10 -1 -2 3 • Graphical 6 • Verbal The cost is twice the original amount. Example • Penney’s is having a sale on coats. The coat is marked down 37% from it’s original price at the cash register. • 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? Function Notation The Symbolic Form • A truly excellent notation. It is concise and useful. y f x y f x Name of the function • Output Value • Member of the Range • Dependent Variable • Input Value • Member of the Domain • Independent Variable These are all equivalent names for the y. These are all equivalent names for the x. Example of Function Notation • The f notation f x x 1 f 2 2 1 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. Average National Price of Gasoline Graphs and Functions Range Domain CBR 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. Determine the Domain and Range for Each Function From Their Graph 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. Numeric • Tables of points are the most common way of representing a function numerically Verbal • Describing the relation in words. We did this with the opening examples. Key Points • Definition of a function • Ways to represent a function Symbolically Graphically Numerically Verbally