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Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,*,*,Definition of a Relation,A,relation,is any set of ordered pairs.The set of all first components of the ordered pairs is called the,domain,of the relation,and the set of all second components is called the,range,of the relation.,Definition of a RelationA rela,Example:,Analyzing U.S.Mobile-Phone Bills as a Relation,Solution,The domain is the set of all first components.Thus,the domain is,1994,2019,2019,2019,2019.,The range is the set of all second components.Thus,the range is,56.21,51.00,47.70,42.78,39.43.,Find the,domain and range,of the relation,(1994,56.21),(2019,51.00),(2019,47.70),(2019,42.78),(2019,39.43),Example:Analyzing U.S.Mobile-,Definition of a Function,A function is a correspondence between two sets,X,and,Y,that assigns to each element,x,of set,X,exactly one element,y,of set,Y,.For each element,x,in,X,the corresponding element,y,in,Y,is called the,value,of the function at,x,.The set,X,is called the,domain,of the function,and the set of all function values,Y,is called the,range,of the function.,Definition of a FunctionA func,Example:,Determining Whether a Relation is a Function,Solution,We begin by making a figure for each relation that shows set X,the domain,and set Y,the range,shown below.,Determine whether each relation is a function.,a.(1,6),(2,6),(3,8),(4,9)b.(6,1),(6,2),(8,3),(9,4),1,2,3,4,6,8,9,Domain,Range,(a),Figure(a)shows that every element in the domain corresponds to exactly one element in the range.No two ordered pairs in the given relation have the same first component and different second components.Thus,the relation is a function.,6,8,9,1,2,3,4,Domain,Range,(b),Figure(b)shows that 6 corresponds to both 1 and 2.This relation is not a function;two ordered pairs have the same first component and different second components.,Example:Determining Whether a,When is a relation a function?,T=(1,2),(3,4),(6,5),(1,5),Note that the first component in the first pair is the same as the first component in the second pair,therefore T is not a function.,Determine whether each relation is a function.,S=(1,2),(3,4),(5,6),(7,8),Each first component is unique,therefore S is a function,When is a relation a function?,Function Notation,When an equation represents a function,the function is often,named,by a letter such as,f,g,h,F,G,or,H,.Any letter can be used to name a function.Suppose that,f,names a function.Think of the domain as the set of the functions inputs and the range as the set of the functions outputs.The input is represented by,x,and the output by,f,(,x,).The special notation,f,(,x,),read,f,of,x,or,f,at,x,represents the value of the function at the number,x,.,If a function is named,f,and,x,represents the independent variable,the notation,f,(,x,)corresponds to the y-value for a given x.Thus,f,(,x,)=4-,x,2,and y=4-,x,2,define the same function.This function may be written as,y=,f,(,x,)=4-,x,2,.,Function NotationWhen an equat,Example:,Evaluating a Function,Solution,We substitute 2,x,+3,and-,x,for,x,in the definition of,f,.When replacing,x,with a variable or an algebraic expression,you might find it helpful to think of the functions equation as,f,(,x,)=,x,2,+3,x,+5.,If,f,(,x,)=,x,2,+3,x,+5,evaluate:,a.,f,(2)b.,f,(,x,+3)c.,f,(-,x,),a.We find,f,(2)by substituting 2 for x in the equation.,f,(,2,)=,2,2,+3,2,+5=4+6+5=15,Thus,f,(,2,)=15.,more,more,Example:Evaluating a Function,Example:,Evaluating a Function,Solution,b.,We find,f,(,x,+3)by substituting,x,+3 for x in the equation.,f,(,x,+,3,)=(,x,+,3,),2,+3(,x,+,3,),+5,If,f,(,x,)=,x,2,+3,x,+5,evaluate:,a.,f,(2)b.,f,(,x,+3)c.,f,(-,x,),Equivalently,f,(,x,+,3)=(,x,+,3,),2,+3(,x,+,3,),+5,=,x,2,+6,x,+9+3,x,+9+5,=,x,2,+9,x,+23.,Square x+3 and distribute 3 throughout the parentheses.,more,more,Example:Evaluating a Function,Example:,Evaluating a Function,Solution,c.,We find,f,(-,x,)by substituting-,x,for x in the equation.,f,(,-,x,)=(,-,x,),2,+3(,-,x,),+5,If,f,(,x,)=,x,2,+3,x,+5,evaluate:,a.,f,(2)b.,f,(,x,+3)c.,f,(-,x,),Equivalently,f,(,-,x,)=(,-,x,),2,+3(,-,x,),+5,=,x,2,3,x,+5.,Example:Evaluating a Function,Finding a Functions Domain,If a function f does not model data or verbal conditions,its domain is the largest set of real numbers for which the value of,f,(,x,)is a real number.,Exclude from a functions domain real numbers that cause,division by zero and,real numbers that result in an even root of a negative number.,Finding a Functions DomainIf,Example:,Finding the Domain of a Function,Solution,Normally it is safe to say the domain of a function is all real numbers.However,there are 2 conditions which must be considered:,1)division by zero,and,2)even roots of negative numbers,.Consider the following functions and find the domain of each function:,a.The function,f,(,x,)=,x,2,7,x,contains neither division nor an even root.,The domain of,f,is the set of all real numbers.,more,more,b.The function contains division.Because division by 0 is und
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