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Functions form 3
 

Functions form 3

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    Functions form 3 Functions form 3 Presentation Transcript

    • What is a FunctionDefinition 1A function is a rule that produces acorrespondence/relation between two sets ofelements, A and B, such that to each element in thefirst set, A, there corresponds one and only oneelement in the second set, B.The first set is called the domain and the secondset is called the co-domain.The set of all corresponding elements in the secondset is called the range of the function.
    • Functions as Mapping Diagrams A f B a 2 b 4 c 6 d 8 Diagram 1 Does Diagram 1 represent a function? Yes, it does. Each element in the domain A is mapped to one and only one element in the co-domain B. NB: Co-domain ={2, 4, 6, 8} Range = {2,4, 6, 8}
    • Functions as Mapping Diagrams A f B a 2 b 4 c 6 d 8 Diagram 2 Does Diagram 2 represent a function? Yes, it does. Each element in the domain A is mapped to one and only one element in the co-domain B. NB: Co-domain ={2, 4, 6, 8} Range = {4, 6, 8} Since the element 2 in B is not mapped onto
    • Functions as Mapping Diagrams A f B a 2 b 4 c 6 d 8 Diagram 3 Does Diagram 3 represent a function? No, it does not. a A is not mapped to any element in B
    • Functions as Mapping Diagrams A f B 2 b 4 c 6 d 8 Diagram 4 Does Diagram 4 represent a function? No, it does not. b A is mapped to more than one element in B
    • Functions as Mapping Diagrams A f B Function Notation a 2 Under the function, f, b 6 is the image of c i.e. f(c)=6 4 Read as ‘f of c is 6’ c 6 8 is the image of d i.e. f(d)=8 d 8 4 is the image of a and b Diagram 1 i.e. f(a)=f(b)=4 Conversely, we can say that c is the pre-image of 6
    • State which of the following mappings represent functions. If the mapping representsa function state the domain, the co-domain and the range of the function. C g D E f F X f Y 1 1 2 1 1 b 2 4 4 2 a 3 9 3 d 4 4 16 X f Y X f Y s h p n i e c a e u k d a t
    • X f Y Can a function from X to Y be created if each element1 1 in Y is to be mapped onto?2 23 3 4
    • E h F Is the mapping h: E →F a function?2 2 If yes state the domain and the range3 4 of the function4 65 No. 8 2 E is not mapped to any Diagram 1 element in F Show Answer
    • C g D Is the mapping g: C →D a function?2 2 If yes state the domain and the range3 of the function 44 6 Yes. Each element in C is mapped to an5 element in D 8 Domain = {2, 3, 4, 5} Range = {2, 4, 6, 8} Diagram 1 Show Answer
    • X g Y Is the mapping g: X →Y a function?a 2 If yes state the domain and the rangeb of the function 4c 6 No.d 8 b X is mapped to more than one element in Y Diagram 1 Show Answer
    • Ordered Pairs A f B f can also be written as a set of ordered 1 2 pairs where 2 4 f={( , ), ( , ), ( , ), ( , ) 3 6 4 8 Diagram 1Definition 2A function is a set of ordered pairs with the property that no two ordered pairs havethe same first component and different second components.The set of all first components in a function is called the domain of the function, andthe set of all second components is called the range.
    • A f B Write the function f:A→B as a seta 1 of ordered pairs.bc 3 f={(a, 1), (b, 1), (c, 3), (d, 3)}d Diagram 2 Show Answer
    • State which of the following sets of ordered pairs represent a function. If it represents a function state the domain and the range of the function. f={(2,, 4), (3, 6), (4, 8), (5, 10)} Show AnswerYes. No 2 ordered pairs have the same 1st component and different 2nd componentsDomain = {2, 3, 4, 5} Range = {4, 6, 8, 10} g={(1, 1), (1, 2), (2,, 2), (3, 3)} Show AnswerNo. The element 1 is mapped to more than 1 element.