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IIT JEE Coaching For Foundation Classesįunctions can be classified according to their images and pre-images relationships.Structural Organisation in Plants and Animals.Let f(x) = 2x+3 Let g(x) = 3x+2 f ○ g R R R g f g(1) f(5) f(g(1))=13 1 g(1)=5 (f ○ g)(1) f(g(x)) = 2(3x+2)+3 = 6x+7ĭoes f(g(x)) = g(f(x))? Let f(x) = 2x+3 Let g(x) = 3x+2 f(g(x)) = 2(3x+2)+3 = 6x+7 g(f(x)) = 3(2x+3)+2 = 6x+11 Function composition is not commutative! Not equal!ġ8 Useful functions Floor: x means take the greatest integer less than or equal to the number Ceiling: x means take the lowest integer greater than or equal to the number round(x) = x+0. Such a function is a one-to-one correspondence, or a bijection 1 2 3 4 a b c dġ2 Identity functions A function such that the image and the pre-image are ALWAYS equal f(x) = 1*x f(x) = x + 0 The domain and the co-domain must be the same setġ3 Inverse functions Let f(x) = 2*x R f R f-1 f(4.3) 4.3 8.6 f-1(8.6)Ĭan we define the inverse of the following functions? An inverse function can ONLY be done defined on a bijection 1 2 3 4 a b c 1 2 3 a b c d What is f-1(2)? Not onto! What is f-1(2)? Not 1-to-1! one-to-one Are the following functions onto, one-to-one, both, or neither? 1 2 3 4 a b c 1 2 3 4 a b c d 1 2 3 4 a b c 1-to-1, not onto Both 1-to-1 and onto Not a valid function 1 2 3 a b c d 1 2 3 4 a b c d Onto, not 1-to-1 Neither 1-to-1 nor ontoġ1 Bijections Consider a function that is both one-to-one and onto: “A function is surjective” A function is an surjection if it is onto Note that there can be multiply used elements in the co-domain 1 2 3 4 a e i o u An onto functionġ0 Onto vs. 1 2 3 4 a e i o u An onto function 1 2 3 4 5 a e i o A function that is not ontoĩ More on onto Surjective is synonymous with onto Onto functions A function is onto if each element in the co-domain is an image of some pre-image Formal definition: A function f is onto if for all y C, there exists x D such that f(x)=y. “A function is injective” A function is an injection if it is one-to-one Note that there can be un-used elements in the co-domain 1 2 3 4 5 a e i o A one-to-one function 1 2 3 4 5 a e i o A one-to-one function 1 2 3 4 5 a e i o A function that is not one-to-oneħ More on one-to-one Injective is synonymous with one-to-one More functions The image of “a” A pre-image of 1 Domain Co-domain A B C D F Ayşe Barış Canan Davut Emine A class grade function 1 2 3 4 5 “a” “bb“ “cccc” “dd” “e” A string length functionĮven more functions Range 1 2 3 4 5 a e i o u Some function… 1 2 3 4 5 “a” “bb“ “cccc” “dd” “e” Not a valid function! Also not a valid function!ĥ Function arithmetic Let f1(x) = 2x Let f2(x) = x2į1+f2 = (f1+f2)(x) = f1(x)+f2(x) = 2x+x2 f1*f2 = (f1*f2)(x) = f1(x)*f2(x) = 2x*x2 = 2x3Ħ One-to-one functions A function is one-to-one if each element in the co-domain has a unique pre-image Formal definition: A function f is one-to-one if f(x) = f(y) implies x = y.
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A function takes an element from a set and maps it to a UNIQUE element in another set f maps R to Z R Z Domain Co-domain f f(4.3) 4.3 4 Pre-image of 4 Image of 4.3