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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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Presentation on theme: "Objectives: 1.To determine if a relation is a function 2.To find the domain and range of a function 3.To evaluate functions."— Presentation transcript:

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

2 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

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

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

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

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

7 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

8 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

9 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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14 Tell whether or not each table represents a function. Give the domain and range of each relationship.

15 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?

16 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)}

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

18 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.

19 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.

20 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

21 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.

22 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.

23 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

24 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

25 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

26 Determine the domain and range of each function.

27 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.

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

29 Determine the domain of each function

30 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

31 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?

32 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.

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

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

35 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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