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BY Sampath Ix class

BY Sampath Ix class. OUR TEAM. M.JYOTHI. FUNCTIONS. M.SUDHAKAR CHETTY. GEORG CANTOR WHODEVELOPED CONCEPT OF THE SET THEORY (FUNCTIONS). a. g. -. gof. +. 1 2 3. f. GEORG CANTOR. Georg Ferdinand Ludwig Philipp Cantor Born: 3 March 1845 in St Petersburg,

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BY Sampath Ix class

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  1. BY Sampath Ix class

  2. OUR TEAM M.JYOTHI FUNCTIONS M.SUDHAKAR CHETTY

  3. GEORG CANTOR WHODEVELOPED CONCEPT OF THE SET THEORY (FUNCTIONS). a g - gof + 1 2 3 f

  4. GEORG CANTOR Georg Ferdinand Ludwig Philipp Cantor Born: 3 March 1845 in St Petersburg, Russia Died: 6 Jan 1918 in Halle, Germany

  5. GEORG CANTOR BORN AT RUSSIA GEORG CANTOR born in St petersburg, RUSSIA

  6. TYPES OF FUNCTION • One to One Function • On to Function • One to One On to • Inverse of a Function • Equal Function • Identity Function • Constant Function • Composite Function

  7. O N E T O O N E • A function f : A B is said to be One to One Function. If no two distinct elements of A have the same image in B. f x a y b c z A B

  8. On to Function f : A B is said to be an On to Function. If f(A) is the image of A equal B that is f is On to Function if every element of B. The Co-domain is the image of at least one element A the domain. f: A B is on to for every x € B there exist at least one x € A such that f(x) = y f(A) = B. B A f a x b y c z d

  9. One to one on to • A function f : A B is said to be a bijection if it is both one to one and on to. A B f X a Y b Z c

  10. INVERSE FUNCTION • If f is a function then the set of ordered pairs obtained by interchanging the first and second coordinates of each order fair in F s called inverse of F. it denoted by F-1 f = { (0,0), (1,1), (2,4), (2,9)……..} f -1 = { (0,0), (1,1), (4,2), (9,2)……..} A B f 1 2 3 X Y z

  11. IDENTITY FUNCTION A function f A→A is said to be an Identity Function on A denoted by IA . f(x) = x A A x x f : A → A

  12. CONSTANT FUNCTION A Function f : A→B is a constant function if there is an element cЄB such that f(x) =c B A f 1 2 3 a b c d

  13. COMPOSITE FUNCTION • Let F:A→B G:B→C be two functions then the composite function of F and G denoted by gof. g f f : A→B g : B→C gof :A→C

  14. GRAPHS OF FUNCTION Eg-1 Eg-2 Eg-3 Eg-4 Eg-5 Eg-6 O Line l cuts the graph TWICE Cuts the graph once IT IS NOT A FUNCTION IT IS A FUNCTION

  15. Ex-1 Let f, g, h be functions defined as follows f(x)=(x+2); g(x)=3x-1; h(x)= 2x show that ho(gof)=(hog)of . {ho[gof](x) {[hog]of}(x) ={h(gof)(x)} =(hog)[f(x)] =h{g[f(x)]} =h{g[f(x)]} =h[g(x+2)] =h[g(x+2)} =h[3(x+2)-1] =h[3(x+2)-1] =h(3x+5) =h(3x+5) =2(3x+5) =2(3x+5) =6x+10 =6x+10 ho(go) = (hog)of

  16. EXERCISES • Sate and define types of functions. • Define Inverse of a function and Inverse function. • Let A={-1,1}. Let the functions f1 and f2 and f3 be from A into A defined as follows: f1(x)=x; f2(x)=x2 ; f3(x)=x3. • Let f(x)=x2+2, g(x)=x2-2, for xЄR , find fog(x), gof(x).

  17. ACKNOWLEDGEMENTS • http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Cantor.html • Micro soft Encarta. • Telugu Academy Text Book - 10th class.

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