Relations and Functions Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required)
To access a customizable version of this book, as well as other interactive content, visit www.ck12.org AUTHORS Brenda Meery Kaitlyn Spong CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FlexBook, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform. Copyright 2015 CK-12 Foundation, www.ck12.org The names CK-12 and CK12 and associated logos and the terms FlexBook and FlexBook Platform (collectively CK-12 Marks ) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License (http://creativecommons.org/ licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the CC License ), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/terms. Printed: January 24, 2015
www.ck12.org Chapter 1. Relations and Functions CHAPTER 1 Relations and Functions Here you will learn about relations, and what makes a relation a function. The following table of values represents data collected by a student in a math class. Does this set of ordered pairs represent a function? x 5 10 15 10 5 0 y 12 25 37 55 72 0 Watch This Khan Academy Functions as Graphs MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/8 Guidance Consider the relationship between two variables. You can think of this relationship in terms of an input/output machine. If there is only one output for every input, you have a function. If not, you have a relation. Relations can have more than one output for every input. A relation is any set of ordered pairs. A function is a set of ordered pairs where there is only one value of y for every value of x. Look at the two tables below. Table A shows a relation that is a function because every x value has only one y value. Table B shows a relation that is not a function because there are two different y values for the x value of 0. 1
www.ck12.org TABLE 1.1: Table A x y 0 4 1 7 2 7 3 6 TABLE 1.2: Table B x y 0 4 0 2 2 6 2 7 When looking at the graph of a relation, you can determine whether or not it is a function using the vertical line test. If a vertical line can be drawn anywhere through the graph such that it intersects the graph more than once, the graph is not function. Example A Determine if the following relation is a function. TABLE 1.3: x y 3.5 3.6 1 1 4 3.6 7.8 7.2 Solution: The relation is a function because there is only one value of y for every value of x. Example B Which of the following graphs represent a function? 2
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www.ck12.org Solution: In order to answer this question, you need to use the vertical line test. A graph represents a function if no vertical line intersects the graph more than once. Let s look at the first graph. Draw a vertical line through the graph. Since the vertical line hit the graph more than once (indicated by the two red dots), the graph does not represent a function. Since the vertical line hit the graph only once (indicated by the one red dot), the graph does represent a function. 4
www.ck12.org Chapter 1. Relations and Functions Since the vertical line hit the graph only once (indicated by the one red dot), the graph does represent a function. Since the vertical line hit the graph more than once (indicated by the three red dots), the graph does not represent a function. Example C Which of the following represent functions? 5
www.ck12.org Solution: a) This is a function because every input has only one output. b) This is not a function because one input (1) has two outputs (2 and 7). c) This is a function because every input has only one output. Concept Problem Revisited x 5 10 15 10 5 0 y 12 25 37 55 72 0 If you look at this table, there are two places where you see the more than one output for a single input. You can conclude that this set of ordered pairs does not represent a function. It is just a relation. Vocabulary Function A function is an example of a relation where there is only one output for every input. In other words, for every value of x, there is only one value for y. Relation A relation is any set of ordered pairs (x,y). A relation can have more than one output for an input. Vertical Line Test The Vertical Line Test is a test for functions. If you can take your pencil and draw a straight vertical line through any part of the graph, and the pencil hits the graph more than once, the graph is not a function. Guided Practice 1. Is the following a representation of a function? Explain. s = {(1,2),(2,2),(3,2),(4,2)} 6
www.ck12.org Chapter 1. Relations and Functions 2. Which of the following relations represent a function? Explain. 3. Which of the following relations represent a function? Explain. a) x 2 4 6 8 10 12 y 3 7 11 15 19 23 7
www.ck12.org b) c) Answers: 1. s = {(1,2),(2,2),(3,2),(4,2)} This is a function because there is one output for every input. In other words, if you think of these points as coordinate points (x,y), there is only one value for y given for every value of x. 2. a) 8
www.ck12.org Chapter 1. Relations and Functions Since the vertical line hit the graph more than once (indicated by the two green circles), the graph does not represent a function. b) Since the vertical line hit the graph only once (indicated by the one green dot), the graph does represent a function. 3. a) x 2 4 6 8 10 12 y 3 7 11 15 19 23 This is a function because there is only one output for a given input. b) 9
www.ck12.org Since the vertical line hit the graph more than once (indicated by the three blue circles), the graph does not represent a function. c) Since the vertical line hit the graph only once (indicated by the one blue dot), the graph does represent a function. Explore More Determine whether or not each relation is a function. Explain your reasoning. 10 1..
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www.ck12.org Chapter 1. Relations and Functions Which of the following relations represent a function? Explain. 6.. 7.. X 2 3 2 5 Y 3 1 5 4 8.. X 4 2 6 1 Y 2 4 3 5 9.. X 1 2 3 4 Y 5 8 5 8 10.. X 6 5 4 3 Y 4 4 4 4 X 2 0 2 4 Y 6 4 4 6 13
www.ck12.org Which of the following relations represent a function? Explain. 11. s = {( 3,3),( 2, 2),( 1, 1),(0,0),(1,1),(2,2),(3,3)} 12. s = {(1,1),(1,2),(1,3),(1,4),(1,5)} 13. s = {(1,1),(2,1),(3,1),(4,1),(5,1)} 14. s = {( 3,9),( 2,4),( 1,1),(1,1),(2,4)} 15. s = {(3, 3),(2, 2),(1, 1),(0,0),( 1,1),( 2,2)} 14