Unit 4 Relations and Functions 4.1 An Overview of Relations and Functions Jan 26 5:56 PM Jan 26 6:25 PM A Relation associates the elements of one set of objects with the elements of another set. Relations can be represented with: Eample 1: Consider the set ears Y and the set of NHL hocke teams H such that Y = {2010, 2011, 2012, 2013, 2014} H = {Chicago Blackhawks, Boston Bruins, Los Angeles Kings}. With words, we can define the relation 1. Words 2. Sets of ordered pairs 3. Arrow Diagrams 4. Tables of Values 5. Graphs 6. Equations As a set of ordered pairs: Note: There are man was to describe a relations using words. Also, man different relations can eist between two sets. Jan 26 6:27 PM Jan 26 7:32 PM This relation in an Arrow diagram: The Stanle cup went to 2010 Bruins 2011 2012 Blackhawks 2013 Kings 2014 Eample 2: Sam Wonders if there is a relationship between the length of a person's ear and the length of their face. Sam samples 5 of her friends and presents the following data. Length of Ear Length of Face Note: In an arrow diagram we use ovals to represent our sets and each arrow represents how elements associate with one another. Jan 26 7:55 PM Jan 26 8:14 PM 1
Sam notices a relation between the data sets. Represent this relation in an arrow diagram: As a graph: As set of ordered pairs: As an equation in two variables: In words: Jan 26 8:26 PM Jan 26 8:38 PM Problem set 1. p. 262 63 3(a), 4, 7(a,b), 8 When describing relations we often speak in terms of a domain and a range. Domain "maps to" Range The domain is the set of first elements in a relation (i.e., a set of " values" or "inputs" in a relation). The range is the set of second elements in a relation (i.e., a set of " values" or "outputs" in a relation). Jan 27 1:26 PM Jan 26 8:41 PM We also speak in terms of independent/dependent variables. Elements in the domain represent the independent variable (which does not depend on the variation of another variable). Identif the independent variable and dependent variable in each of the following contets. Eample 1: The amount of time spent studing affects the grade a student receives. Elements in the range represent the dependent variable (which depends on the variation of another variable). Eample 2: The temperature outside affects the amount of ice cream sold. Jan 27 10:43 AM Jan 27 12:13 PM 2
A function is a special tpe of relation where each element in the domain is paired with eactl one element in the range. Eample: the following relation is a function. Robert Downe Jr. Chris Evans Edward Norton Mark Ruffalo Scarlett Johansson Acts as Iron Man Black Widow The Hulk Captain America Not all relations are functions. is plaed b Iron Man Black Widow The Hulk Captain America Robert Downe Jr. Chris Evans Edward Norton Mark Ruffalo Scarlett Johansson But all functions are relations! Jan 27 12:23 PM Jan 27 12:28 PM Which of the following relations are functions, which are not? Which of the following relations are functions, which are not? 4 2 3 1 6 4 2 5 1 8 9 4 {(2,4), (3,6), (4,6), (5,6), (6,12)} {( 3,7), (0, 10), (3,13), (3, 5), (6,16)} 1 2 3 7 0 4 3 4 2 Jan 27 1:01 PM Jan 27 1:01 PM How to determine whether a relation is a function: 1. If an values in the domain are associated with more than one value in the range, the relation is not a function. Problem set 2 p. 270 4., 8., 9., 12. 2. A graph that fails the "vertical line test" does not represent a function. Jan 27 1:20 PM Jan 27 1:25 PM 3
A function can be viewed as a rule or a machine that takes input values () and produces output values (). 4.2 Function Notation Equations that represent functions, such as = 4 + 3, ma written using functional notation f(). f() = 4 + 3 Note: This notation can be thought of as another wa of representing the value. Jan 27 1:43 PM Jan 27 8:09 PM f () Eample 1. Write the following in functional notation. a) = 2 + 5 b) v = 5t 2 c) p = n 2/3 Eample 1. Write the following as an equation in two variables. d) f() = 3 + 2 e) h(t) = 6t + 9 Jan 27 8:55 PM Jan 27 8:58 PM Eample 2: if p(r) = 2r + 3 determine the following. a) p(2) b) p( 3) Eample 3. If v(t) = 4t 7 find the value of t when v(t) = 29. Check our answer. NOTE: An input value and an output value f () can be written as ordered pair (, f()). Jan 27 9:02 PM Jan 27 9:37 PM 4
Tom is eating popcorn at the movies. The equation P = 25m + 250 represents the amount of popcorn Tom has left after m minutes. a). Describe the function. Write the equation in functional notation. b). Determine the value of P(3). What does this value represent? Two Minute Write! Give an eample of a an relation (other than the ones discussed in class), and eplain wh or wh not our eample is a function. Creativit is appreciated. c). Determine the value for m when P(m) = 0. What does value this represent? Jan 27 9:41 PM Feb 3 6:19 PM Describing Graphs 4.3 Graphing and Interpreting Data Jan 27 11:19 PM Jan 31 1:15 PM On a distance time graph what do the following represent? 1. A line going upwards to the right. 2. A line going down to the right Sketching Graphs (given a situation) A salmon is swimming upstream. For the first hour, it swims strongl and at a constant speed. For the net two hours, due to fatigue and a stronger current, it swims half as fast. It finall realizes it has passed its destination so, for the final half hour, it stops swimming and allows the current to carr it slowl back downstream. Sketch a possible distance versus time graph to best represent the path of the salmon. 3. A horizontal line Jan 31 3:16 PM Jan 31 2:28 PM 5
Speed time Graphs Feb 7 1:33 PM Feb 7 1:27 PM On a speed time graph what do the following represent? 1. A line going upwards to the right. 2. A line going down to the right Another Sketch A a parked car speeds up at a green light, reaching a constant speed of 50 km/h over a 30 second period. After 5 minutes of traveling at this speed, the car approaches a red light and slows to a stop over the course of 5 seconds. Sketch a possible speed versus time graph to represent the path of the car. 3. A horizontal line Jan 31 3:19 PM Feb 7 1:35 PM The following containers are filled with water at a constant rate. For each container, sketch a graph to represent the height of the water over time. Feb 7 1:33 PM Feb 7 2:11 PM 6
Some things to note when describing and sketching graphs: Problem set 3 p.281 83. 3, 7, 11, 13, 18. 1. Does the line pass through the origin 2. What is the scale of the and aes? 3. What do these aes represent? 4. Where does the data start and end? 5. Is the data continuous or discrete? REMINDER: Do not forget about the other problem sets (1 and 2) or the worksheet (4.2) from last class. Jan 31 2:53 PM Jan 31 2:36 PM Discrete Vs Continuous Data Discrete data involves distinct, countable elements. Jan 31 2:28 PM Jan 31 3:54 PM Epressing Domain and Range. Recall: The domain is the set of all possible values for the independent variable in a relation. The range is the set of all possible values for the dependent variable given the domain. Continuous data can take on an real value (within a given range) Jan 31 3:55 PM Domain and range ma be represented using 1. Words 2. A list 3. A real number line 4. Interval notation 5. Set notation. Jan 31 2:08 PM 7
Consider the following graph. Consider the following graph What is the domain and range in words and on a number line? What is the domain and range in words, as a list and on a number line? Wh not use a list? Is this a function? Jan 31 2:08 PM Jan 31 3:13 PM Interval notation. Smbols are used to indicate an interval of the set of real numbers. This notation uses: [ if the end number is included ( if the end number is not included Use interval notation to epress the domain and range. U When more than one interval is needed if there is no end point Note: Discrete data should not be represented using interval notation. Jan 31 2:08 PM Jan 31 8:11 PM Set notation. The formal and standard wa to represent sets. Set notation generall looks like { < 3, A } Use set notation to epress the domain and range. "The set of" "all " "such that" "is an element of" Note: 1. the statement < 3 ma be an statement. 2. "A" man be an set such as W, N, I, Q, or R. Jan 31 2:33 PM Jan 31 8:11 PM 8
Some more eamples Number line Interval Set Notation Notation 4.4 Linear Relations Jan 31 2:34 PM Feb 2 7:38 PM E 1. Which of the following graphs are linear? Linear relationships are relations of direct proportionalit, or of constant change. In other words, a relation is called linear if an change in an independent variable results in a corresponding (or constant) change in the dependent variable. When plotted on a graph, a linear relation traces a straight line, hence the name. Feb 2 7:47 PM Note: For linear relations, a constant change can be observed in both the independent and the dependent variable. Feb 2 8:02 PM E 2. Which of the following tables of values represent linear relations? E 2. Which of the following sets of ordered pairs represent linear relations? f() p() g() a) {(1, 2), (1, 1), (1, 0), (1, 1), (1, 2)} 2 15 4 30 6 45 8 60 10 75 1 1 2 4 3 9 4 16 5 25 6 5 3 5 0 5 3 5 6 5 b) {(0, 3), (1, 6), (2, 9), (3, 12), (4, 15)} c) {( 2, 8), ( 1, 1), (0, 0), (1, 1), (2, 8)} Feb 2 8:08 PM Feb 2 8:43 PM 9
Eploration: For each of the following functions: i) Generate a list of ordered pairs or a table [HINT: start with the input values 2, 1, 0, 1, and 2, then etend our graph if necessar]. ii) Graph the function and determine whether the relation is linear 1. f() = 2 + 3 1. f() = 2 + 3 2. g(t) = t 2 t 1 Feb 2 8:08 PM Feb 2 9:07 PM 1. g(t) = t 2 + t 1 In Sum, we can determine if a relation is linear b 1. Checking if the graph traces a straight line 2. Determining if a constant change in the independent variable results in a constant change in the dependent variable. 3. Observing the degree of the equation (linear relations are of degree 1). Problem set 4. Pg. 308. 3, 4, 5, 6 (i, iii, v). Feb 2 9:07 PM Feb 2 9:28 PM 4.5 Characteristics of Linear Relations In this last section we will eamine the following characteristics of linear relations: 1. Rate of change 2. Intercepts 3. Domain 4. Range Feb 4 6:41 PM Feb 4 7:02 PM 10
The Rate of Change of a Linear Relation is constant and can be determined b observing changes in the independent and dependent variable. As a graph: Consider the following relation: The cost of a tai is $3.50, plus $.75 for ever km traveled. Cost ($) Distance (km) Cost ($) 0 3.50 1 4.25 2 5.00 3 5.75 4 6.50 Feb 3 6:35 PM Distance (km) Feb 4 7:27 PM Eample: The cost for a car rental is $60, plus $20 for ever 100 km driven. What is the rate of change in the cost per kilometer? Some notes about rate of change. 1. The steeper the line, the greater of the rate of change. 2. The sign of a rate of change (whether it is positive or negative), determines the direction of the line. Side Question: What is the rate of change for each of the given graphs? Feb 4 7:54 PM Feb 4 7:37 PM Intercepts of a linear relation. A horizontal (or ) intercept is a point where a line crosses the ais. at these points = 0, so, an intercept can be represented as an ordered pair Eamples. What are the and intercets for each of the following relations? A vertical (or ) intercept is a point where a line crosses the ais. at these points = 0, so, an intercept can be represented as an ordered pair Feb 3 6:36 PM Feb 4 9:11 PM 11
Finding and intercepts, given a linear relation. To find a intercept, let = 0 and solve for [i.e., solve f(0)]. To find the intercept, let = 0 and solve for [i.e., let f() = 0]. Challenge Time! Sketch a linear relation with Eactl 2 intercepts Eactl 1 intercept E: Find the and intercepts for = 2 + 3. An infinite number of intercepts Feb 5 5:49 PM Feb 4 9:21 PM Intercepts can have important meanings in contetual problems. State the Domain and Range for each Linear Relation. Note: Vertical intercepts often correspond to starting points and horizontal intercepts often correspond to ending points. Feb 4 9:29 PM Feb 4 9:27 PM Contetual problems This graph shows Mitchell leaving home at point A and going to a part at point F. Two Minute Write: Write a scenario for the following graph What was the slowest speed? What about the ma? Which point represents when he turned around to go back home. What might be a reason for this? When did Mitchell stop? Wh? Feb 3 6:43 PM Feb 5 12:17 PM 12
Dominoe's Pizza charges $11.99 for a 12'' Hand Tossed pizza (this includes onl cheese and pizza sauce). The cost per etra topping is $1.50. Write an equation in two variables to represent the relation between the number of etra toppings and the cost of a pizza. What is the rate of change? What does it represent? What is the vertical intercept? What does it represent? Graph the relation, assuming that ou ma have a maimum of 10 etra toppings. Is there an intercept? eplain. What is the Domain and Range for the relation? Feb 5 12:22 PM Feb 5 5:54 PM We ma sketch a graph of a linear relation if ou know onl two points (sa, the and intercept). Eample: Sketch g() = 4 + 12. If we onl know the rate of change and a single point, we can also sketch a graph for a linear relation. E: A large hot tub contains about 1500 liters of water and is being drained at a constant rate of 100 liters per minute. Sketch a graph that represents the amount of water in the hot tub as a function of time. Note: Be careful to determine if our data is continuous or discrete. If no contet is provided, assume that it is continuous. Feb 4 10:08 PM Feb 5 6:37 PM Problem set 5. Page 308 09. 7, 14 Page 319 22. 4, 5, 8, 16, 17 Feb 4 10:01 PM Jan 26 11:35 AM 13