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Objectives: 1.To determine if a relation is a function 2.To find the domain and range of a function 3.To evaluate functions

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As a class, use your vast mathematical knowledge to define each of these words without the aid of your textbook. RelationFunction InputOutput DomainRange Set-Builder Notation Interval Notation Function Notation

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relation A mathematical relation is the pairing up (mapping) of inputs and outputs.

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Domain Domain: the set of all input values Range Range: the set of all output values

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function A toaster is an example of a function. You put in bread, the toaster performs a toasting function, and out pops toasted bread.

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What comes out of a toaster? It depends on what you put in. – You cant input bread and expect a waffle!

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function A function is a relation in which each input has exactly one output. A function is a dependent relation Output depends on the input Relations Functions

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function A function is a relation in which each input has exactly one output. Each output does not necessarily have only one input Relations Functions

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BIG If you think of the inputs as boys and the output as girls, then a function occurs when each boy has only one girlfriend. Otherwise the boy gets in BIG trouble. Darth Vadar as a Procurer.

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Tell whether or not each table represents a function. Give the domain and range of each relationship.

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cardinality The size of a set is called its cardinality. What must be true about the cardinalities of the domain and range of any function?

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Which sets of ordered pairs represent functions? 1.{(1, 2), (2, 3), (3, 4), (3, 5), (5, 6)} 2.{(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)} 3.{(1, 1), (2, 1), (3, 1), (4, 1), (5, 1)} 4.{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5)}

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Which of the following graphs represent functions? What is an easy way to tell that each input has only one output?

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A relation is a function iff no vertical line intersects the graph of the relation at more than one point Function Not a Function If it does, then an input has more than one output.

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To determine if an equation represents a function, try solving the thing for y. Make sure that there is only one value of y for every value of x.

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Determine whether each equation represents y as a function of x. 1. x 2 +2 y = 4 2.( x + 3) 2 + ( y – 5) 2 = 36

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set-builder notation Since the domain or range of a function is often an infinite set of values, it is often convenient to represent your answers in set-builder notation. Examples: { x | x < -2} reads the set of all x such that x is less than negative 2.

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set-builder notation Since the domain or range of a function is often an infinite set of values, it is often convenient to represent your answers in set-builder notation. Examples: { x : x < -2} reads the set of all x such that x is less than negative 2.

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interval notation Another way to describe an infinite set of numbers is with interval notation. Parenthesis not Parenthesis indicate that first or last number is not in the set: – Example: (-, -2) means the same thing as x < -2 – Neither the negative infinity or the negative 2 are included in the interval – Always write the smaller number, bigger number

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interval notation Another way to describe an infinite set of numbers is with interval notation. Brackets Brackets indicate that first or last number is in the set: – Example: (-, -2] means the same thing as x -2 – Infinity (positive or negative) never gets a bracket – Always write the smaller number, bigger number

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Domain Domain: All x - values (L R) – { x : - < x < } Range Range: All y - values (D U) – { y : y -4} Domain: All real numbers Range: Greater than or equal to -4

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Determine the domain and range of each function.

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Domain Domain: What you are allowed to plug in for x. – Easier to ask what you cant plug in for x. – Limited by division by zero or negative even roots – Can be explicit or implied Range Range: What you can get out for y using the domain. – Easier to ask what you cant get for y.

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Determine the domain of each function. 1. y = x

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Determine the domain of each function

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depends Functions can also be thought of as dependent relationships. In a function, the value of the output depends on the value of the input. Independent quantity Independent quantity: Input values, x -values, domain Dependent quantity Dependent quantity: Output value, which depends on the input value, y -values, range

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The number of pretzels, p, that can be packaged in a box with a volume of V cubic units is given by the equation p = 45 V In this relationship, which is the dependent variable?

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In an equation, the dependent variable is usually represented as f ( x ). Read f of x – f = name of function; x = independent variable – Takes place of y: y = f (x) – f ( x ) does NOT mean multiplication! – f (3) means the function evaluated at 3 where you plug 3 in for x.

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Evaluate each function when x = f ( x ) = -2 x g ( x ) = 12 – 8 x

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Let g ( x ) = - x x + 1. Find each function value. 1. g (2) 2. g ( t ) 3. g ( t + 2)

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Objectives: 1.To determine if a relation is a function 2.To find the domain and range of a function 3.To evaluate functions Assignment: Continue Pgs #47-79 odd

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