Language of Relations and Functions PDF

The Language of Relations and Functions Relation n A relation is a correspondence between a first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to at least one member of the range.  Example 1: What makes this a relati...

The Language of Relations and Functions Relation n A relation is a correspondence between a first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to at least one member of the range.  Example 1: What makes this a relation? {(0, 5), (1, 4), ( 2, 3), ( 3, 2), (4, 1), (5, 0)} What is the domain? {0, 1, 2, 3, 4, 5} What is the range? {-5, -4, -3, -2, -1, 0} Unit 4 - Relations and Functions 3 Example 2 – Is this a relation? Input 4 –5 0 9 –1 Output –2 7 What is the domain? {4, -5, 0, 9, -1} What is the range? {-2, 7} Unit 4 - Relations and Functions 4  Is a relation a function? What is a function? According to a textbook, “a function is…a relation in which every input has exactly one output” 5  Is a relation a function? Focus on the x-coordinates, when given a relation If the set of ordered pairs has different x-coordinates, it IS A function same x-coordinates, If the set of ordered pairs has it is NOT a function Y-coordinates have no bearing in determining functions 6  Example 3 {(0, 5), (1, 4), ( 2, 3), ( 3, 2), (4, 1), (5, 0)} Is this a relation? YES Is this a function? Hint: Look only at the x-coordinates YES 7  Example 4 {(–1, 7), (1, 0), ( 2, 3), (0, 8), (0, 5), (–2, 1)} Is this a function? Hint: Look only at the x-coordinates NO Is this still a relation? YES 8 Example 5 Which relation mapping represents a function? Choice One Choice Two 3 –1 2 2 1 2 –1 3 0 3 3 –2 0 Choice 1 9 Example 6 Which relation mapping represents a function? A. B. B 10  Vertical Line Test Vertical Line Test: a relation is a function if a vertical line drawn through its graph, passes through only one point. 11 Vertical Line Test Would this graph be a function? YES 12 Vertical Line Test Would this graph be a function? NO 13 Is the following function discrete or continuous? What is the Domain? What is the Range? Discrete -7, 1, 5, 7, 8, 10    1, 0, -7, 5, 2, 8    14 Is the following function discrete or continuous? What is the Domain? What is the Range? continuous   8,8     6,6   15 Is the following function discrete or continuous? What is the Domain? What is the Range? continuous   0,45    10,70   16 Is the following function discrete or continuous? What is the Domain? What is the Range? discrete   -7, -5, -3, -1, 1, 3, 5, 7   2, 3, 4, 5, 7 17  Example 7 Which situation represents a function? a. The items in a store to their prices on a certain date b. Types of fruits to their colors There is only one price for each A fruit, such as an apple, from the different item on a certain date. The domain would be associated with relation from items to price makes it a more than one color, such as red and function. green. The relation from types of fruits to their colors is not a function. 18  Domain and Range in Real Life The number of shoes in x pairs of shoes can be expressed by the equation y = 2x. What is the independent variable? The # of pairs of shoes. What is the dependent variable? The total # of shoes. 19  Domain and Range in Real Life Mr. Landry is driving to his hometown. It takes four hours to get there. The distance he travels at any time, t, is represented by the function d = 55t (his average speed is 55mph. What is the independent variable? The time that he drives. What is the dependent variable? The total distance traveled. Unit 4 - Relations and Functions 20  Domain and Range in Real Life Johnny bought at most 10 tickets to a concert for him and his friends. The cost of each ticket was $12.50. Complete the table below to list the possible domain and range. 1 2 3 4 5 6 7 8 9 10 12.50 25.00 37.50 50 62.50 75 87.50 100 112.50 125 What is the independent variable? The number of tickets bought. What is the dependent variable? The total cost of the tickets. Unit 4 - Relations and Functions 21  Domain and Range in Real Life Pete’s Pizza Parlor charges $5 for a large pizza with no toppings. They charge an additional $1.50 for each of their 5 specialty toppings (tax is included in the price). What is the independent variable? The number of toppings What is the dependent variable? The cost of the pizza Unit 4 - Relations and Functions 22 Operation of Functions Review: What is a function? l A relationship where every domain (x value has exactly one unique range (y value). l Sometimes we talk about a FUNCTION MACHINE, where a rule is applied to each input of x Function Operations Addition :  f  g ( x)  f x   g x  Multiplica tion :  f  g  x   f  x   g  x  Subtraction :  f  g x   f x   g x  f f x  Division :   x   where gx   0 g g x  Adding and Subtracting Functions Let f  x   3 x  8 and g  x   2 x  12. Find f  g and f - g  f  g ( x)  f x   g x  f  g ( x)  f  x   g  x   (3x  8)  (2 x  12)  (3 x  8)  (2 x  12)  5x  4  x  20 When we look at functions we also want to look at their domains (valid x values). In this case, the domain is all real numbers. Multiplying Functions Let f  x   x - 1 and g x   x  1. 2 Find f  g f x   g ( x)  ( x 2  1)( x  1) In this case, the domain is all real numbers because  x3  x 2  x  1 there are no values that will make the function invalid. Dividing Functions Let f  x   x 2 - 1 and g x   x  1. f  Find   g f x  x 2  1   g x  x  1 In this case, the domain is all real numbers EXCEPT -1, because x=-1 would give ( x  1)( x  1) a zero in the denominator.  x 1 ( x  1)