![]() 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. 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
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