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Functions Exercises

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  Injective, Surjective and Bijective Injective, Surjective and Bijective tell you about how a function behaves. A function is a way of matching the members of a set A to  a set B : A General Function  points from each member of A to a member of B . To be a function you never  have one A pointing to more than one B , so one-to-many is not OK  in a function (as you would have something like f(x) = 7 or  9 ) But more than one A can point to the same B ( many-to-one is OK )  Injective  means that every member of A has its own unique  matching member in B . As it is also a function  one-to-many is not OK  And you won't get two A s pointing to the same B , so many-to-one is NOT OK . But you can have a B without a matching A Injective functions can be reversed ! If A goes to a unique B then given that B value you can go back again to A (this would not work if two or more A s pointed to one B like in the General Function ) Read Inverse Functions for more. Injective is also called One-to-One Surjective  means that every B has at least one  matching A (maybe more than one). There won't be a B left out. Bijective  means both Injective and Surjective together.  So there is a perfect one-to-one correspondence between the members of the sets. (But don't get that confused with the term One-to-One used to mean injective). On The Graph Let me show you on a graph what a General Function and a Injective Function looks like: General Function Injective (one-to-one) In fact you can do a Horizontal Line Test : To be Injective , a Horizontal Line should never intersect the curve at 2 or more points. (Note: Strictly Increasing (and Strictly Decreasing) functions are Injective, you might like to read about them for more details)    Formal Definitions OK, stand by for some details about all this: Injective   A function f   is injective  if and only if whenever f(x) = f(y) ,  x = y  . Example:   f  (  x  ) =  x+5   from the set of real numbers to is an injective function. This function can be easily reversed. for example:    f(3) = 8  Given 8 we can go back to 3 Example:   f  (  x  ) =  x  2  from the set of real numbers to is not  an injective function because of this kind of thing:    f  ( 2 ) = 4 and    f  ( -2 ) = 4  This is against the definition f(x) = f(y) ,  x = y  , because f(2) = f(- 2) but 2 ≠ -2  

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Jul 23, 2017

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