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Copyright © 2016 Education, Inc. Section R.2 Slide R- 2 Copyright © 2016 Education, Inc. FUNCTION: A function 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 exactly one member of the range. R.2 Functions and Models Slide R- 3 Copyright © 2016 Education, Inc. a) Microsoft Stock Domain Range March 14, 2006 $27.23 March 15, 2006 $27.36 March 16, 2006 $27.27 March 17, 2006 $27.50 Example 1: Determine whether or not each correspondence is a function. This relationship is a function because each member of the domain corresponds to only one member of the range. R.2 Functions and Models Slide R- 4 Copyright © 2016 Education, Inc. Example 1 (continued): b) Squaring Domain Range 3 9 4 16 5 –5 25 This relationship is a function because each member of the domain corresponds to only one member of the range, even though two members of the domain correspond to 25. R.2 Functions and Models Slide R- 5 Copyright © 2016 Education, Inc. Example 1 (continued): c) Baseball Teams Domain Range Arizona Diamondbacks Chicago Cubs White Sox Baltimore Orioles This relationship is not function because one member of the domain, Chicago, corresponds to two members of the range, Cubs and White Sox. R.2 Functions and Models Slide R- 6 Copyright © 2016 Education, Inc. Example 1 (continued) : d) Baseball Teams Domain Range Diamondbacks Arizona Cubs Chicago White Sox Orioles Baltimore This relationship is a function because each member of the domain corresponds to only one member of the range, even though two members of the domain correspond to Chicago. R.2 Functions and Models Slide R- 7 Copyright © 2016 Education, Inc. Example 2: The squaring function, f , is given by Find R.2 Functions and Models Slide R- 8 Copyright © 2016 Education, Inc. A function is given by . Find: In Class Question Slide R- 9 Copyright © 2016 Education, Inc. Example 3: A function subtracts the square of an input from the input. A description of f is given by Find R.2 Functions and Models Slide R- 10 Copyright © 2016 Education, Inc. Example 3 (concluded): R.2 Functions and Models Slide R- 11 Copyright © 2016 Education, Inc. A function is given by Find: In Class Question Definition The graph of a function f is a drawing that represents all the input-output pairs, (x, f (x)). In cases where the function is given by an equation, the graph of a function is the graph of the equation y = f (x). Slide R- 12 Copyright © 2016 Education, Inc. R.2 Functions and Models Slide R- 13 Copyright © 2016 Education, Inc. Example 4: Graph f (x) = x2 –1. R.2 Functions and Models The Vertical Line Test A graph represents a function if it is impossible to draw a vertical line that intersects the graph more than once. Slide R- 14 Copyright © 2016 Education, Inc. R.2 Functions and Models Slide R- 15 Copyright © 2016 Education, Inc. Example 4: Determine whether each of the following is the graph of a function. R.2 Functions and Models Slide R- 16 Copyright © 2016 Education, Inc. Example 4 (concluded): a) The graph is that of a function. It impossible to draw a vertical line that intersects the graph more than once. b) The graph is not that of a function. A vertical line (in fact many) can intersect the graph more than once. c) The graph is not that of a function. d) The graph is that of a function. R.2 Functions and Models Slide R- 17 Copyright © 2016 Education, Inc. Example 5: In 2005, Sprint® offered a cellphone calling plan in which a customer’s monthly bill can be modeled by the graph below. The amount of the bill is a function f of the number of minutes of phone use. R.2 Functions and Models Slide R- 18 Copyright © 2016 Education, Inc. Example 5 (continued): Each open dot on the graph indicates that the point at that location is not included in the graph. a) Under this plan, if a customer uses the phone for 360 min, what is his or her monthly bill? b) If a monthly bill is $55, for how many minutes did the customer use the phone? R.2 Functions and Models Slide R- 19 Copyright © 2016 Education, Inc. Example 5 (concluded): a) To find the bill for 360 min of use, we locate 360 on the horizontal axis and move directly up to the graph. We then move across to the vertical axis. Thus, the bill is $40. b) To find the number of minutes of use when a monthly bill is $55, we locate 55 on the vertical axis, move horizontally to the graph, and note that many inputs correspond to 55. If t represents the number of minutes of use, we must have 600 < t ? 700. R.2 Functions and Models Slide R- 20 Copyright © 2016 Education, Inc. Example 6: Graph the function defined as follows: The function is defined such that g(1) = 3 and for all other x-values (that is, for x ? 1), we have g(x) = –x + 2. Thus, to graph this function, we graph the line given by g(x) = –x + 2, but with an open dot at the point above x = 1. To complete the graph, we plot the point (1, 3) since g(1) = 3. R.2 Functions and Models Slide R- 21 Copyright © 2016 Education, Inc. Example 6 (concluded): R.2 Functions and Models Slide R- 22 Copyright © 2016 Education, Inc. Extra Example : Graph the function defined as follows: