MA 15200 Lesson 20 Section 2.1A relation is any set of ordered pairs. The set of all first components of the ordered pairsis called the domain. The set of all second components is called the range.Relations can be represented by tables, sets, equations of two variables, or graphs.Name % of all NamesSmith 1.006%Johnson 0.810%Williams 0.699%Brown 0.621%Jones 0.621%Ex 1: Find the domain and range of each relation.a){(2, 3),(3, 4),(4, 5),(5, 5),(2, 6)}- - - - -A relation in which each member of the domain corresponds to exactly one member of the range is a function. The table above that pairs a last name with a percent of all namesis a function because each last name is paired to exactly one percent. Another way to identify a function is the following, it is a relation in which no two ordered pairs have the same first component and different second components.Definition of a FunctionA function is a correspondence from a first set, called the domain, to a second set, calledthe range, such that each element in the domain corresponds to exactly one element in the range.If the table was changed as below, the relation is not a function. The percent 0.621% would be paired with both Brown and Jones.% of all Names Name1.006% Smith0.810% Johnson0.699% Williams0.621% Brown0.621% Jones1The table at the left would represent a relation where the ordered pairs are of the form (name, %).An example would be (Williams, 0.699%). The domain would be {Smith, Johnson, Williams, Brown, Jones} and the range is (1.006%, 0.810%, 0.699%, 0.621%}.I Determining Whether a Relation is a FunctionEx 2: Determine if each relation is a function. a) x y0 11 0-1 02 -3-3 -8-2 -34 -15b){(2, 3),(3, 4),(4, 5),(5, 5),(2, 6)}- - - - -II Determining Whether an Equation Represents a FunctionMany functions are written as equations of two variables. For example, 0.6 94R x=- +, where R represents the average number of meals per person Americans ordered from restaurants and x represents the number of years after 1984. The x is called the independent variable, because any number of years after 1984 can be selected. The y iscalled the dependent variable, because its value depends upon the value of x.Not all equations represent functions. If an equation is solved for y and more than one value of y can be obtained for a given x, then the equation does not define a function.Ex 3: Determine if each equation is a function or not. Find the domain in interval notation.2 2) 2 c) 2) 3 d) 1a y x x yb y x xy y= + = += - - =-22) 5xe yx=-III Determining if a Graph Represents a FunctionGraphs of Functions: A function can be graphed by determining the set of all ordered pairs (points) where x is in the domain and y is in the range. Because each x can only be paired to one y, the vertical line test can be used to determine if a graph represents a function. If every possible vertical line would intersect the graph only once, then the graph represents a function.Ex 2: Determine which graphs are functions.3The Vertical Line Test for FunctionsIf any vertical line intersects a graph in more than one point, the graph does not define y as a function of x.IV Function NotationFunctions are often named using letters such as f, g, h, F, G, or p. The input of the function is x. The output of the function is y, which can be represented as f(x), read f of x or function of x. This notation means the value of the function at the number x and is known as function notation.The notation f(2) means replace a 2 for the value of x (or the independent variable) in the function. Some substitutions for x do not have to be numbers. f(x-4) says to find the function value when the original x is replaced with x – 4. Remember f(x) means the same thing as y. The function value is the y or f(x) value.Ex 3: If 24)( and 23)(2 xxgxxxf, find the following.2) ( 1)) ( 4)) ( )) (2 )) ( 2)) ( 1)) ( 5)a fb gc f ad g ke g mf f ng g x- =- ===- =+ =+ =V Graphs of FunctionsThe graph of a function is the graph of its ordered pairs. We have already graphed some equations, but will graph one function here.4Ex 4: Graph ( ) 2 3g x x= - The graph above is a line. Any function of the form ( )f x mx b= + has a straight line graph and is called a linear function.VI Analyzing the Graph of a FunctionEx 5: For each graph shown, find the following.a) The Domain in set-builder notation and interval notation.b) The Range in set-builder notation and interval notation.c)(3)fd) For what value of x is ( ) 4f x =?e) Any x-interceptsf) Any y-intercept5VII Applied Function ProblemsThe bar graph below shows the average final exam score for 4 consecutive years.62004 2005 2006 2007Number of Points70100806068717376The functions ( ) 3 56f x x= + and 22( ) 565g x x= +model the average exam score where x is the number of years after 2000.Ex 6: a) According to the graph, what was the average exam score in 2007?b) Using the function f model, what was the average exam score in 2007?c) Does function f underestimate or overestimate the actual exam score according to the graph? By how much?d) Using the function g model, what was the average exam score in 2007?e) Does the function g overestimate or underestimate the actual average given in the graph? By how
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